UC-NRLF 


^B    531    fll5 


V 


\i,  c-.t 


)}v(i'- I  '■■) 


)■)'■■■  -  ;.i 

■,(li'.i5KA 


LIBRARY 

OF  THE 

University  of  California. 

GIFT    OF 

Class 

I 


I 


APPLETOJ^S'  MATHEMATICAL  SERIES 


NUMBERS  UNIYERSALIZED 

AN 

ADVANCED  ALGEBRA 


BY 

DAVID   M.  SENSENia,  M.S. 

PROFESSOR  OF  MATHEMATICS,   STATE  NORMAL  SCHOOL,  WEST  CHESTER,   PA. 


PART  SECOND 


NEW  YORK,  BOSTON,  AND  CHICAGO 

D.    APPLETON    AND    COMPANY 
1890 


Copyright,  1890, 
By  D.  APPLETON  AND  COMPANY. 


PEEFAOE 


NuMBEKS  Universalized  is  believed  to  embrace  all 
algebraic  subjects  usually  taught  in  the  preparatory  and 
scientific  schools  and  colleges  of  this  country.  For  con- 
venience, it  is  divided  into  two  parts,  which  are  bound  sepa- 
rately and  together,  to  accommodate  all  kinds  and  grades 
of  schools  sufficiently  advanced  to  adopt  its  use. 

Part  Second  is  treated  in  five  chapters,  as  follows  :  One 
embracing  serial  functions,  including  development  of  func- 
tions into  series,  convergency  and  divergency  of  infinite 
series,  the  binomial  formula,  the  binomial  theorem,  the 
exponential  and  logarithmic  series,  summation  of  series, 
reversion  of  series,  recurring  series,  and  decomposition  of 
rational  fractional  functions  ;  one  treating  of  complex  num- 
bers, graphically  and  analytically,  including  fundamental 
operations  with  complex  numbers,  general  principles  of 
modulii  and  norms,  and  the  development  and  representa- 
tion of  sine,  cosine,  and  tangent ;  one  embodying  a  discus- 
sion on  the  theory  of  functions,  including  graphical  repre- 
sentations of  the  meaning  of  the  terms  independent  and 
dependent  variables,  continuous  and  discontinuous  func- 
tions, increasing  and  decreasing  functions,  and  turning 
values  and  limits  of  functions,  and  also  a  treatment  of 
differentials  and  derivatives,  and  maxima  and  minima  val- 
ues of  functions ;  one  treating  of  the  theory  of  equations, 
including  a  discussion  of  the  properties  of  the  roots,  real 
and  imaginary,  of  an  equation,  methods  of  determining 
the  commensurable  roots  of  a  numerical  equation,  Sturm's 
theorem  for  detecting  the  number  and  situation  of  real 
roots,  Horner's  method   of  root  extension.  Cardan's  for- 

183681 


iv  PREFACE. 

mnla  for  solving  cubic  equations,  and  a  short  treatment 
of  reciprocal  and  binomial  equations ;  one  treating  of  de- 
terminants and  probabilities,  so  far  as  these  subjects  are  of 
interest  and  value  to  the  general  student.  The  volume 
closes  with  a  supplementary  discussion  of  continued  frac- 
tions and  theory  of  numbers. 

The  aim  of  the  author  in  preparing  this  part  of  his  work 
has  not  been  so  much  to  give  completeness  to  the  various 
subjects  treated  as  to  lead  the  student  to  a  comprehension 
of  the  fundamentals  of  a  wider  range  of  subjects,  and  to 
cultivate  in  him  a  taste  for  mathematical  investigation. 
It  is  believed  that  the  plan  adopted  will  give  the  general 
student  a  broader  and  more  practical  knowledge  of  algebra, 
and  will  lead  to  better  results  in  a  preparatory  course  of 
study  for  the  university  than  would  a  completer  treatment 
of  fewer  subjects  requiring  an  equal  amount  of  space  in 
their  development  and  more  time  in  their  mastery.  While 
a  sufficient  number  of  examples  have  been  placed  under 
each  head  to  offer  opportunity  for  the  application  of  the 
principles  and  laws  developed,  there  will  not  be  found  an 
unnecessary  multiplicity  of  them  to  retard  the  progress  of 
the  pupil  in  his  onward  course. 

In  conclusion,  the  author  desires  to  acknowledge  his  in- 
debtedness to  the  English  authors,  Hall  and  Knight,  Chr^s- 
tal,  Aldis,  Whitworth,  and  0.  S.  Smith,  whose  works  he 
frequently  consulted,  and  from  which  he  obtained  many 
new  and  valuable  ideas. 

David  M.  Sensei^ig. 

Normal  School,  West  Chester,  Pa.,  \ 

December  2,  1889,  \  • 


OOIfTElsrTS, 


CHAPTER  IX. 

PAGE 

Serial  Functions 315-353 

Definitions,  315. 

Development  of  Functions  into  Series 816-319 

Theorem  of  indeterminate  coefficients,  316.  Expansion  of  rational 
fractions,  316-318.    Expansion  of  irrational  functions,  318. 

Convergency  and  Divergency  of  Infinite  Series       .        .        .  319-323 
General  definitions,  319.    Fundamental  principles,  320.    Theorems, 
321,  322.    Exercises,  323. 

The  Binomial  Formula 324-326 

Development  of,  324.    Applications,  325,  336. 

The  Binomial  Theorem 326-332 

For  positive  exponents,  326,  327.    For  any  rational  exponents,  328- 

)         =  nr«— I,  328.    General  demonstration  of 

binomial  theorem,  328-330.    Numerous  corollaries  and  inferences, 
327,  330-332.    Exercises,  332. 

The  Exponential  Theorem 333 

The  Logarithmic  Series 334-337 

Development  of,  334,  335.  Computation  of  logarithms,  335,  336. 
Principles,  336,  337. 

Summation  of  Series 337-344 

Method  by  indeterminate  coefficients,  338.  Method  by  decompo- 
sition, 339.  The  differential  method,  339-342.  To  find  the  (n  +  l)th 
term  of  a  series,  340.  To  find  the  sum  of  n  terms,  341.  To  inter- 
polate terms,  342.    Exercises,  343. 

Reversion  of  Series    .        . 344 

Recurring  Series 345-349 

Definitions,  345.  To  determine  the  scale  of  coefficients,  346.  To  find 
the  snm  of  n  terms,  347.    Exercises,  349. 

Decomposition  of  Rational  Fractional  Functions    .        .        .  350-353 
Definition,  350.    Principles,  350-352.    Exercises,  353. 


vi  CONTENTS. 

CHAPTER  X. 

PAGE 

Complex  Numbers 354-365 

Graphical  Treatment 354-361 

Definitions,  354-358.    To  add  complex  numbers  graphically,  359.    To 

multiply  a  complex  number  by  a  rational  number,  graphically,  360. 

By  a  simple  imaginary  number,  360.    By  a  complex  number,  361. 

General  Principles  of  Complex  Numbers  .....  361-362 

Problem.    To  find  the  value  of  e^+y* 363 

Graphical  Representation  of  sin.  y  and  cos.  y  ,        .        .        .  364-365 

CHAPTER  XI. 
Theory  of  Functions 366-388 

Definitions,  366-370. 

Graphical  Representation  of  Functions 370-374 

Exercises,  374. 

Differentials  and  Derivatives  of  Functions      ....  374-382 

Definitions,  374.  Principles,  375-379.  Exercises,  380-383.  Concrete 
applications,  380.    Successive  derivatives,  381. 

Factorization  of  Polynomials  containing  Equal  Factors .        .  382-383 
Maxima  and  Minima  of  Functions 385-388 

CHAPTER  XII. 

Theory  of  Equations 389-422 

Introduction,  389.  Normal  forms,  390,  391.  Divisibility  of  equations, 
393.  Number  of  roots,  393.  Relation  of  roots  to  coefllcients,  394. 
Imaginary  roots,  395.  Fractional  roots,  396.  Relation  of  roots  to 
signs  of  equations,  396.  Descartes'  rule  of  signs,  398.  Limits  of  ■ 
roots,  399.  Equal  roots,  400.  Commensurable  roots,  400-403.  In- 
commensurable roots,  403-409.  Sturm's  series  of  functions  and 
fundamental  principles,  404-406.  Sturm's  theorem,  406-409.  Hor- 
ner's method  of  root  extension,  409-415.  Cubic  equations,  415-419. 
Cardan's  formula,  416-419.  Recurring  equations,  419-422.  Bino- 
mial equations,  422. 

CHAPTER  XIII. 

Determinants 423-441 

Introduction,  423-425.  Properties  of,  425-430.  Development  of,  430- 
432.  Additional  properties  of,  432-435.  Multiplication  of  deter- 
minants, 435-437.  Solution  of  simultaneous  equations  by  determi- 
nants, 438-440.  Conditions  of  simultaneity,  440,  441.  Sylvester's 
method  of  elimination,  441. 


CONTENTS.  vii 

PAGE 

Probabilities 443-453 

Definitions  and  fundamental  principles,  442-444.  Exclusive  events, 
444-446.  Expectation,  446.  Independent  events,  447,  448.  Inverse 
probability,  448-450.  Probability  of  testimony,  450,  451.  Exer- 
cises, 451-453. 


SUPPLEMENT. 
Continued  Fractions 454-463 

Definitions,  454,  455.  Formative  law  of  successive  convergents,  455- 
457.  Properties  of  convergents,  457-460.  Keduction  of  common 
fractions  to  continued  fractions,  460,  461.  Reduction  of  quadratic 
surds  to  continued  fractions,  461,  462.  Reduction  of  periodic  con- 
tinued fractions  to  simple  fractions,  462.  Approximation  to  the 
ratio  of  two  numbers,  462.    Exercises,  463. 

Theory  of  Numbers 464-482 

Systems  of  notation,  464^i67.  Divisibility  of  numbers  and  their 
digits,  468-470.  Even  and  odd  numbers,  470-473.  Prime  and  com- 
posite numbers,  473-477.    Perfect  squares,  477^79.    Perfect  cubes, 

479,  480.    Exercises,  480-482 


PAET    SECOND. 


CHAPTER   IX. 
SERIAL    FUJVCTIOJTS. 


I.   Definitions. 


662.  Any  expression  containing  a  variable  is  called  a 
function  of  the  variable. 

Thus,  ax -{-!?,  a x~\  Va-\-a^,  a",  log.  {a  -\-  x),  and 
a-{-l)x-\-cx^-{-doi?-\-  etc. ,  are  functions  of  x. 

663.  Any  series  containing  variable  terms  is  called  a 
serial  function, 

664.  The  expression  f{x)  represents  any  function  of  x, 
and  is  read  function  x, 

666.  When  two  or  more  functions  of  the  same  variable 
are  used  in  a  discussion,  modified  forms  are  used  for  dis- 
tinction ;  as, 

1.  f{x),  F{x),  <f>(x);  read,  /  minor,  f  major,  phi 
functions  of  x. 

^'  f  (^).  /"  i^\  f"  (^) ;  read,  /  prime,  f  second, 
f  third  functions  of  x. 

3.  /i  (x),  /2  (x),  fs  (x) ;  read,  /  one,  f  two,  f  three 
functions  of  x. 


316  ADVANCED  ALGEBRA, 

Development  of  Functions  into  Series. 

Theorem  of  Indeterminate  Coefficients. 

566.  If  A-\-Bx-{-Ca^-\-D^etc.  :=  A^-^- B^x-{-C^x^ 
-\- DiO^-\-  etc.,  for  any  assigned  value  of  x  from  —  cx)  to 
+  00,  atid  A,  At,  B,  B^,  C,  C^,  D,  D^,  etc.,  are  independ- 
ent of  X,  then  will  A  =  A^,  B  =  B^,  C=^Ci,  D~  D^  etc. 

Demonstration:  Given 
A->c  Bx^-  Cx^  -{-Dq?  +  etc.  =  A^  +  B^x  +  CiX^  +  D^a^  +  etc.,  (A) 
for  any  assigned  value  of  x.    Let  a;  =  0 ;  then  A  =  Ai. 

Therefore,  A  =  Ai  for  every  value  of  x.  (1) 

Subtract  (1)  from  (A), 

Bx  +  Gx^  ^Da?-^  etc.  =  BiX  ^t  CxX^  +  D^a?  -\-  etc.  (B) 

Divide  (B)  by  cc, 

B  ^Cx  +  Dxy^  ■\-  etc.  =  ^1  +  (7i a;  +  Di  a;2  +  etc.  (C) 

Let  a;  =  0,  5  =  B^. 

Therefore,  B  =  Bi  for  every  value  of  x.  (2) 

etc.,  etc.,  etc. 

567.  Corollary  l.  —  If  A  +  Bx -^  Cx^  +  Dx^ -{-  etc. 
=  0,  for  any  assigned  value  of  x,  then  will  A=0,  B  =  0, 
(7=0,  D  =  0,  etc. 

568.  Cor.  2. — A  function  of  a  single  variaUe  can  he  de- 
veloped into  a  series  of  the  ascending  powers  of  the  variable 
in  only  one  way. 

For,  if  possible,  let 

f{x)  =  a-\-'hx-\-co?  -\-  etc. ;  and 
f{x)  =  «i  +  Jj  a;  +  Ci  x^  +  etc. ;  then  will 
a-^-lx-^cx^ -\-  etc.  =  ax-^-lyX-^-c^o? -\-  etc. ; 
whence  a  =  ai,  J  =  Ji,  c  =  c^,  etc.,  and  the  two  develop- 
ments will  be  identical.    - 

2.  Applications. 
1.  Expansion  of  Rational  Fractions. 

569.  A  rational  fraction  of  a  single  variable  may  gen- 
erally be  developed  into  a  series  by  dividing  the  numerator 


EXPANSION  OF  RATIONAL  FRACTIONS.       317 

by  the  denominator,  but  a  more  expeditious  method  con- 
sists in  the  application  of  the  principle  of  indeterminate 

coefficients. 

1  —  x 
Ulustrations. — 1.  Develop  ^  into  a  series  of  the 

ascending  powers  of  x. 

Let  \^  =  A  +  Bx  +  Cx^  +  Da^  +  etc.  (A) 

1.  •{•  X 

Clear  of  fractions,  and  arrange  the  coefficients  of  the  like  powers 
of  X  into  columns. 


1-x  =  A  +  B 
+  A 


x  +  C 
+  B 


x^  +  D  I  x^+  etc.  (B) 

+  C 


Equate  the  coefficients  of  the  like  powers  of  x  [566], 

A  =  l',  A  +  B  =  -1,  B  +  G  =  0;  C  +  D  =  0,  etc. 
/.       A  =  1,  B  =  -2,  C  =  2,  D  =  -2,  etc. 
Substitute  these  values  of  the  coefficients  in  (A), 

2-^  =  1  -  2a;  +  2a;2  -  2a;8  +  etc. 
1  +  a; 

Let  the  student  divide  the  numerator  by  the  denominator,  and 
show  that  the  same  result  will  follow. 


570.  The  first  term  of  the  expansion  may  be  obtained 
by  dividing  the  first  term  of  the  numerator  by  the  first 
term  of  the  denominator,  and  the  remaining  terms  by 
indeterminate  coefficients. 

2.  Develop  ,    „  in  the  ascending  powers  of  x. 

X    —J~    0   Xr 

^^^      A   2  =  »^~*  +  B^  +  Cic  +  Dx^  +  etc.  (A) 

X  "T  0  X 

Clear  of  fractions  and  column  coefficients, 


x^+  D 
+  Cb 


ic«  +  etc.  (B) 


a  =  a  +  B   I  x+  C 
+  ab  \      +  Bb 
Equate  coefficients, 
(!)  B  +  ab  =  0.        (2)  G  +  Bb  =  0.        (3)  i)  +  C&  =  0,  etc. 
B  =  —  ab,   G  =  ab\  D  =  -  ab\  etc. 
Substitute  these  values  in  (A), 

J— ^  =  aa;-* —  aft  +  a6'a;  —  a62a;' +  etc. 

a;  +  &a;* 


318 


ADVANCED  ALGEBRA, 


EXERCISE    87. 

Develop  to  four  terms  : 

2x  —  S 

x-\-x^  +  l 


2. 


3. 


1-x 
1 


2x 

1 


l-x-\-x^ 


5. 


9. 


a-\-x 
6x  +  2x^ 

l-\-x-{-a^ 


2.  Expansion  of  Irrational  Functions. 


Elnstrations. — 1.  Expand  to  four  terms  Vl—x-\-  x^, 
^-  1  +  Bx  +  Cx^  +  Do?  +  etc. 


Put  Vl  —  a:  +  a;' 

Square  both  members  and  column  the  coefficients, 


Equate  the  coefficients, 


+  3(7 


a;8+  2i) 
+  3^(7 


7?  +  etc. 


(A) 

(B) 


(1)3^=- 
B^- 


(2)  ^  +  3(7=  1. 
3 


(3)  32>  +  3^C  =  0. 


^-8 


D  = 


W 


etc. 


Substitute  these  values  in  (A) 
1 


2a;  +  |a;«+^a:«+etc. 


-s/l  -  a;  +  ^2  =  1 

2.  Expand  to  three  terms  v8  — ^. 

Put  V8-a;2  =  2  +  ^a;  +  Cx^  +  Z>a:3  +  ^a;*  +  etc. 

Cube  both  members  and  column  coefficients. 


8-a;8  =  8  +  13^  I  a;  +  13  C 
'     +    6^ 


a; 

2  +  J53 

+  13i> 

+  13^(7 

a:3  +  13^ 
+    3J52(7 
+    6(7« 
+  13^i> 


Equate  the  coefficients, 

(1)  135  =  0.  (3)  13  C  +  6-B2  =  -  1. 

(3)  135C+13i>  +  53  =  0. 
.    (4)  12jE^+3J52C7+6C8  +  135i>  =  0. 


...     5  =  0,   C=-l,  i> 

Substitute  these  values  in  (A), 


0,  ^=  -So5»  etc. 


V8 


^  =  ^-lV-38-8^-^^- 


(A) 


xl^  +  etc. 


CONVERGENCY  OF  INFINITE  SERIES.         319 

EXERCISE    88. 

Expand  to  four  terms  : 


1.  a/4  — a;  4.  Vl  +  a;  7.  Va-{-x 

2.  ^l^X'-x'  5.  a/27 +  0,-2  8.  Vo^^ 

3.  V9  +  a;-3a;2         6.  Vs  +  M^        9.  VoH^ 


Convergency  and  Divergency  of  Infinite  Series. 

General  Definitions. 

671.  The  limit  of  a  series  is  the  limit  of  the  sum  of  n 
terms  of  the  series,  when  n  is  indefinitely  increased  ;  that 
is,  when  lim.  n=  co, 

672.  A  series  is  convergent  when  its  limit  is  a  finite 
constant,  including  zero, 

673.  A  series  is  divergent  when  its  limit  is  infinity. 

674.  A  series  is  indeterminate  when  the  sum  of  n  terms 
is  finite  hut  does  not  approach  any  definite  yalue  as  n  is 
indefinitely  increased. 

Thus,  1  —  1  +  1  —  1  +  1  —  1  + is  indeterminate, 

since,  when  n  is  even  the  sum  is  0,  and  when  n  is  odd 
the  sum  is  1,  however  great  n  he  taken. 

676.  For  convenience  of  discussion,  the  following  nota- 
tion will  be  adopted : 

1.  The  terms  of  a  series  will  be  represented  in  order 
by  Wi,  Uz,  Us w„,  Un+i 

2.  The  sum  of  n  terms  will  be  represented  by  Z7„,  so 
that  Un  =  Ui-{-U2-\-Us-{- +  w«.  . 

3.  The  limit  of  the  series  will  be  represented  by  U,  so 
that  [7=  wi  +  t^2  +  %  +  ....  +  «^n  +  w«+i  + 


320  ADVANCED  ALGEBRA. 

Fundamental  Principles. 

576.  X  No  series  whose  terms  are  all  of  the  same  sign 
can  he  indeterminate. 

For  either  the  sum  of  n  terms  increases  numerically 
without  limit  as  n  is  increased  indefinitely,  or  else  it  can 
never  exceed  some  fixed  value  which  it  approaches  as  a 
limit.  Such  a  series  is,  therefore,  either  convergent  or 
divergent. 

577.  2,  A  series  of  finite  terfns  whose  signs  are  all 
alike  is  divergent. 

For,  if  we  let  a  represent  the  numerical  value  of  the 
smallest  term,  then,  numerically,  U>na,  whose  limit 
is  c» ,  when  lim.  n  =  co  and  a  is  a  finite  quantity. 

Thus,  the  series  1  +  2  +  44-8  +  16  +  ....  is  divergent. 

678.  3.  If  a  series  is  convergent  it  will  remain  con- 
vergent, and  if  divergent  it  will  remain  divergent,  if  any 
finite  number  of  terms  he  added  to  or  subtracted  from 
the  series. 

For,  the  sum  of  any  finite  number  of  terms  is  finite, 
and,  therefore,  can  not  change  the  nature  of  the  limit  of 
the  series  when  combined  with  the  series  by  addition  or 
subtraction. 

579.  Ji..  If  a  series  is  convergent  when  its  terms  are 
all  positive,  it  is  also  convergent  when  its  terms  are  all 
negative,  or  some  positive  and  some  negative. 

For  its  limit  will  have  the  same  numerical  value  when 
its  terms  are  all  negative  as  when  they  are  all  positive, 
and  will  be  numerically  less  when  the  terms  do  not  all 
have  the  same  sign  as  when  they  do. 

It  must  not  be  inferred  from  this  principle  that  a  series 


THEOREMS.  321 

is  necessarily  divergent  when  its  terms  are  not  all  of  the 
same  sign,  if  it  is  divergent  wh«n  they  are  alike  in  sign. 
Such  may  or  may  not  be  the  case. 


Theorems. 


580.  I.  In  order  that  a  series  may  he  convergent,  the 
limit  of  the  {n  +  l)th  term,  and  the  limit  of  the  sum  of 

.  any  numder  of  terms  beginning  with  the  {n  +  ^)th  term 

must  he  zero,  and  conversely. 

DemonBtration :  If  a  series  is  convergent,  then  ultimately,  if  ti  is 
indefinitely  increased, 

(1)  U-Un      =  o    [4981  (2)  U-  Un+l  =  o 

(3)  U-  Un+^  =  o  (4)  U-  Un+z  =  o 

Subtract  (1)  from  (2) ;  (1)  from  (3) ;  (1)  from  (4),  etc. ;  then, 
(a)  Un  —  Un  +  \  =  o  ;  or,  Un+\  =  o  ;  whence,  lim.  w^^  i  =  0 
(6)   Un  —  Un+i  =  o  ;  or,  Un+\  +  w„4.8  =  o  ;  whence, 

lim.  (Wn+l  +  Wn+j)  =  0 
(c)   Un  —  Un  +  3  =  o  ;  or,  Un  +  1  +  Un  +  i  +  Un+z  =  o  ;  whencc, 

lim.  (Un  +  l  +  Un  +  1  +  Un  +  z)  =  0 

etc.,  etc.,  etc.,  etc. 

581.  II.  If  each  term  of  a  series  whose  terms  are  alter- 
nately positive  and  negative  is  numerically  greater  than 
the  following  term,  the  series  is  convergent. 

Demonstration  :  Let  U=ui  —  Ui  +  U9  —  Ui+ ±  w«  T  Un+i. . . ., 

in  which  -Mi  >  'Wa  >  Ws  >  '^4. . . .,  be  the  given  series. 

(1)  U  —  {Ui  —  Wa)  +  (Ws  —  Ui)  +  (We  —  w«)  +  etc. 

(2)  U  =  Ui  —  (Ui  —  Ws)  —  {Ui  —  -Mb)  —  etc. 
From  (1)  it  is  evident  that  U  is  positive. 

From  (2)  it  is  evident  that,  since  U  is  positive,  U<,Ui. 

.'.  U  approaches  Wi  or  some  quantity  less  than  Ui  as  a  limit, 
and  the  series  is,  therefore,  convergent. 

582.  III.  A  series  is  convergent  if  after  some  particu- 
lar  term  the  ratio  of  each  term  to  the  preceding  term  is  less 
than  unity. 


322  ADVANCED  ALGEBRA. 

Demonstration:  The  most  unfavorable  case  to  convergency  sup- 
posable,  under  the  conditions  given,  is  evidently  the  one  in  which  all 
the  terms  have  the  same  sign  (say  plus)  and  all  the  ratios  described 
are  equal  and  each  equal  to  the  greatest  of  them.  This  is,  therefore, 
the  only  case  that  needs  proof. 

Let  r  be  the  greatest  ratio  after  the  nth  term,  but  <  1 ;  then, 

Un  +  ^n  +  l  +  t^n  +  2  +  ^n  +  S  +   etC.    =    W„  +  W»  7*  +  Wn  r«  +    CtC.    =    q— ^?— 

1  —  r 
[499,  P.]  =  a  finite  quantity.  Therefore,  the  whole  series  is  con- 
vergent [578J. 

683.  IV.  A  series  of  all  positive  or  all  negative  terms 
is  divergent,  if  after  some  particidar  term  the  ratio  of 
each  term  to  the  preceding  term  is  equal  to  or  greater 
than  unity. 

Demonstration :  The  most  unfavorable  case  to  divergency,  and  the 
only  one  that  needs  investigation,  is  the  one  in  which  all  the  ratios 
described  are  equal  and  each  equal  to  the  least  of  them. 

Let  r  be  the  least  ratio  after  the  nth  term,  but  =  or  >  1 ;  then, 
«*n  +  w«+i  +  -Wn+s  +  'Wn+s  +  ctc.  is  divergent  [577]  ;  and  hence  the 
whole  series  is  divergent  [578]. 

584.  V.  A  series  of  positive  terms  is  convergent  if 
each  term  is  less  than  the  corresponding  term  of  a  given 
convergent  series  of  positive  terms. 

Demonstration :  Let  CT  =  Wi  +  w,  +....+  w„  +  u^^i  + be  a 

given  convergent  series ;  and  T  =  Vi  +  Va  + +  Vn  +  Vn+\  + . . . . 

a  series  in  which  Vi  <  Wi ,  iJa  <  "Wa, . . . .  t^n  <  w„ ,  VnJ^\  <  w„+i , 

From  the  nature  of  addition,  it  is  evident  that  Vn  <  Cn ;  and 
hence,  too,  lim.  F„  <  lim.  C/« ,  or  V  <U',  therefore,  if  U  is  converg- 
ent V  is  convergent. 

585.  The  foregoing  principles  and  theorems  will  serve 

to  test  the  convergency  and  divergency  of  a  very  large 

number  of  series,  but  are  not  of  universal  application, 

inasmuch  as  they  do  not  apply  to  all  classes  of  series. 

Note. — If  the  terms  of  a  series  are  not  all  of  the  same  sign  no  gen- 
eral method  can  be  obtained  for  testing  their  convergency  or  divergency. 

586.  The  convergency  or  divergency  of  a  series  may 
often  be  determined  by  grouping  terms,  as  follows : 


THEOREMS.  323 


1     1     i     i     i 


=  1  +  (^3  +  p)  +  (^3  +  5^3  +  ^3  +  fa)  +  (p  +••••+  1^3)  +  etc. 

•••  C^  <  1  +  I  +  ^8  +  I  +  etc.;  or  CA  < J  =  3  . 

,'.  The  series  is  convergent. 


EXERCISE    89. 

1.  Is  the  series  T+o  +  o+T  +  "i'  +  ^^^'  convergent  ? 

1        /C        o        4        0 

2.  Test  the  series  :  l-\-^x-\-hoi?-{-^3?-\- for  con- 

vergency 

1.  When  x<l.      2.  When  x>l.      3.  When  x  =  l. 

3.  Testtheseries:i-^  +  ^-^+.... 
for  convergency. 

when  a;  <  1  ? 

6.  Testtheseries:i  +  ^  +  ^  +  ^  +  etc. 
for  convergency. 

6.  Test  the  series  : 

.    1     rr^       1     3    ^5       1  .  3  .  5    ^.7 

^+2-3+2T4-5+2TT:^-T  +  '*'- 
for  convergency 

1.  When  x<l.      2.  When  x>l.      3.  When  ic  =  1. 
Suggestion.— Lim.  jj  "    =  -j.    Why! 

7.  Test   the   series :    l  +  aj-f-a^^  +  a^^H-   etc.    for  con- 
vergency 

1.  When  x  =  l.      2.  When  x<  1.      3.  When  x>l, 

8.  Test   1  +  1  +  ^  +  ^^^+....    for  con- 

vergency. 


324  ADVANCED  ALGEBRA. 

The  Binomial  Formula. 

587.  The  binomial  formula  is  used  to  find  the  prod- 
uct of  any  number  of  binomial  functions  of  the  form  of 

Development. 
{x  +  a){x  +  'b)  =  x^  +  (a+b)x+ab 
Multiply  both  members  hj  x  +  c, 
{x  +  a){x+b){x  +  c)  =  x^+(a  +  b)x^  +  abx  +  cx^  +  (ac  +  bc)x+abc 
=  x^  +  {a+b  +  c)x^  +  {ab  +  ac  +  bc)x       +abc 
Multiply  both  members  hj  x  +  d, 
{x  +  a)  (x  +  b)  {x+ c)  (x  +  d)  = 
a^+{a+b  +  c)x^  +  {ab  +  ac  +  bc)x^  +  abcx 

+  dx^  +  {ad  +  bd+cd)x'^  +  {abd  +  acd  +  bcd)x+abcd 
=  a^+{a  +  b  +  c  +  d)a^  +  {ab  +  ac  +  ad  +  bc  +  bd+cd)x^  + 

{abc  +  abd  +  acd  +  bcd)x  +  abcd 

Observe  the  following  laws  in  these  products  : 

i.  The  number  of  terms  is  one  greater  than  the  num- 
ber of  binomial  factors, 

2.  The  exponent  of  x  in  the  first  term  equals  the  num- 
ber of  binomial  factors,  and  decreases  by  unity  in  each 
succeeding  term, 

S,  The  coefficient  in  the  first  term  is  unity ;  in  the 
second  term  the  sum  of  the  second  terms  of  the  binomial 
factors  ;  in  the  third  term  the  sum  of  the  products  of  the 
second  terms  taken  two  together  j  in  the  fourth  term  the 
sum  of  the  products  of  the  second  terms  taken  three  to- 
gether, etc, 

4'  The  last  term  equals  the  product  of  all  the  second 
terms. 

Are  these  laws  true  for  any  number  of  factors  ? 

Assume  them  true  for  r  factors,  so  that 

{x-{-a){x-\-b) {x-\-m)  =  x^-^pi  of-^-\-pz  af-*+ .... 

Pr-\  x-\-p^,  in  which 


THE  BINOMIAL  FORMULA.  325 

p^  =  a-\-h-{- +m 

p^  =  al)-\-ac-{- am-\-hc  +  bd-{- +  Jm  +  etc. 

^3  =  abc-\-aI)d-\- +  «5m  +  .... 

p,  =  abc m.  (A) 

Multiply  by  {x  +  n),  the  (r  +  l)th  factor,  then 

{x-\-a){x-{-b)..,.(x  +  n)  = 

X'  +  ^+PlX'-\-      ^2^-^  +  ....+      Pr^ 

-\-  nx'  +  npi  iC"- ^  +  ....  +  npr-i  x       +  npr 

=:af+-'-^{pi-\-n)3f  +  (p2-{-np^)af-''-\-....  +  npr 

Laws  1  and  2  are  evidently  still  true. 

Pi-\-n      =  a -\- h -\- c -{- n, 

Vi  +  ^  A  =  {d^ -\- d  0 -\- am-\-'bc-\-hd-\-.,,. 

•\-lm-\-  etc.)  -\-{an-\-ln-{- -^-mn), 

which  is  still  the  product  of  the  second  terms  taken  two 

and  two. 

•  ••••• 

np,  =  alcd n.     Therefore,  all  the  laws  still  hold 

true.  Hence,  if  they  are  true  for  r  factors,  they  are  true 
for  r  + 1  factors.  But  we  found  them  true  for  four  fac- 
tors by  multiplication  ;  hence,  they  are  true  for  five  fac- 
tors ;  and,  if  so,  for  six  factors ;  and  so  on.  Therefore, 
formula  (A)  is  general. 

Uote. — The  number  of  products  that  enter  into  each  coeflBcient 
may  be  determined  by  the  principles  of  combination. 

Applications. 
Ulnstrations. — 

1.  Expand  {x  + 1)  (^  +  2)  {x  -  3)  (a;  +  4). 

Solution : 
pi  =  1  +  2-3  +  4  =  4 

i?a  =  (1  X  2)  +  (1  X  -3)  +  (1  X  4)  +  (2  X  -3)  +  (2  x  4)  +  (-3  x  4)  =  -7 
i)3  =  (lx2x-3)  +  (lx2x4)  +  (lx-3x4)  +  (2x-3x4)  =  -34 
i>4  =  lx2x— 3x4=  —24 

/.  (x+l)(a;+2)(a;-3)(a;+4)  =  ic4  +  4ic»-7a;»-34a;-24. 


326  ADVANCED  ALGEBRA. 

2.  Factor  a;*  +  14  a;^  _|.  ^^  ^2  _|_  I54  ^  _l_  ^go,  if  possible. 

Let  {x  +  a){x  +  h){x  +  c){x+d)  =  o^  +  l^a?  +  1l\x^  +  l^^x  +  \20. 
Then,  1.  a-\-h  +  c  +  d  =  +14: 

2.  ab  +  ac  +  ad+bc  +  bd  +  cd  =  +71 

3.  a&c  +  a6c^  +  «C(^  +  6ccZ  =  +154 

4.  aJc6?  =  +120 

Resolve  if  possible  +120  into  four  factors  whose  sum  is  +14. 
These  we  find  to  be  2,  3,  4,  5. 

.*.  a  =  2,  b  =  d,  c  =  4,  and  d  =  5. 
Will  these  values  satisfy  2  and  3  ? 

ab  +  ac  +  ad+bc  +  bd  +  cd  =  6  +  8 +  10  +  12  +  15  +  20  =  71,  correct. 
abc  +  abd  +  acd  +  bcd  =  24:+S0  +  40  +  60  =  154,  correct. 

.-.  x^  +  Ux^  +  71x^  +  154:X  +  120  =  (x+2)(x+d)(x  +  4)ix+n). 

EXERCISE    90. 

1.  Expand  (x  -{-2){x-\-  3)  (^  +  1) 

2.  Expand  (x  -j-3)(x  —  2)  (x  —  3) 

3.  Expand  {x  +  2)  (ir  +  3)  {x  -  1)  {x  -  2) 

4.  Expand  (:r  +  3)  (:r  +  5)  (a;  -  2)  (:c  -  6) 

5.  Expand  {x  +  2)  (rr  +  2)  {x  -\-%){x-\-  2) 

6.  Expand  {x  —  h){x  —  5)  (ic  —  5)  (a;  —  5) 

7.  Expand  (2a;  + 1)  (2a;  + 3)  (2:r  -  5)  (2  a;- 1) 
Suggestion. — Put  y  for  2  a:. 

8.  Factor  a;^  _j_  9  ^^^s  _{_  26  a;  +  24 

9.  Factor  x^ -%x^  -%^x-\-m 

10.  Factor  a:*  +  5  a;^  +  5  a;^  —  5  a;  —  6 

11.  Factor  it-*  -  2  a;^  -  25  a;^  +  36  a;  +  120 

12.  Factor  a;^  +  4  a;*  -  13  a;^  -  52  ar^  +  36  a;  +  144 


The  Binomial  Theorem. 

I .  For  Positive  Exponents. 

588.  If,  in  the  binomial  formula  [587,  A],  we  assume 
a  =  h=:c  =  dy  etc.,  and  r  =  w,  then  will 


TEE  BINOMIAL   THEOREM.  327 

1.  {x^a){x  +  h){x-{-c)....  ={x^ay. 

2.  a;'"  =  cc"  ;  x'-^  =  aj~-^ ;  a:'- ^  =  x""-^ ;  etc. 

3.  p^z=a-\-a-\-a-{- to  n  terms  =  na. 

4.  j92  =  «^  +  «^  +  ^^  + =  ^^  taken  as  many  times 

as  there  are  combinations  of  2  in  7i ;  or, 

n(n  —  l)    2 

5.  p^  =  fl^aa  +  aaa  +  «a«  + =  a^  taken   as  many 

times  as  there  are  combinations  of  3  in  w ;  or, 

Ps  =  -^ i ^' 

•  •••••• 

6.  p'=aXaXaXa to  n  factors  =  a\ 

589.  Cor,  1, — If  a  and  x  le  interchanged,  {a  +  xy  = 
ar^na--^x-{-'^^^^a^-^x^-^...,-\-x\  (C) 

From  (B)  and  (C)  it  will  be  seen  that  the  coefficients  of 
any  two  terms  equidistant  from  the  first  and  last  terms 
are  numerically  equal. 

590.  Cor,  2, — If  X  le  made  negative  in  (C),  (a  —  xf  = 

a-  -na''-^x-\-  ^^^"^^  a'^-^x^  -,...±x\ 

591.  Cor.  3, — The  sum  of  the  coefficients  in  (C)  equals 
zero. 

For,  put  a  =  1  and  a:  =  1 ;  then 

Li  Li 


ADVANCED  ALGEBRA. 

2.  For  any  Rational  Exponents. 

— — — J       =  Tir*"^  for  any 

Demonstration :  I.  Let  n  =  any  positive  integer. 

Now,  — ^-^  =  ic"-^  +  ra;«-2  +  r^a;"-^  + +  r«-J  [134]. 

I         =:  lim.  a:"— ^  +  lim.  roc:^—^  + 

x-r  Jx  =  r 

+  lim.  r»-i  [401,  413]  =  r^-^  +  r»->  +  r«-i  + . . . . 

to  n  terms  =  nr^—'^. 

If) 

II.  Ze^  w,  =  i- ,  a  positive  fraction. 

-T        a?^  —  r"^      Xq  —  rq  .. 

Now,  =  — -.  (1) 

'    a;  — r  x  —  r  ^  ' 

Put  Xq  =y,  or  x  =  y9  '^  and  r?  =  s,  or  r  =  s? ;  then 
a;»  —  r**       yp  —  sp       yp  —  sp       y^  —  s? 
X  —  r    ~  yt  —  si  ~    y  —  s         y  —  8  ' 
Since  x-=yi  and  r  =  s^,  lim.  y  =  s  when  lim.  ic  =  r. 

.,  Lim.  (^^^:r.\        =  ita.  -S ^^^1^'  ^  ^1^' i        = 
\aJ  —  r/x  =  r  i  y  —  s         y  —  s)y  =  8 

pyP-i  ^  qyq--^  [I,  416]  =  ^  yp-<i  = 

p     J_  p—<i       p     P.  _i 

—  (xq)        =  —  xg        =^na^—'^. 

III.  Lei  n  =  —p,  a  negative  integer  or  fraction. 

__        x*  —  r*       x—P  —  r-P  /a;p  —  rP\ 

Now,  = =  —  x-P  r-P  ( I . 

'    x  —  r  x  —  r  \  x  —  r  J 

VaJ  — r/a:  =  r  |  \x  —  rj\x  =  r 

—  —r-^p,prP-^  [415]  =  —  r'^«(— ?ir-»*-')  =  nr»-i. 

General   Dennonstratlon  of  BInonnial  Theorem. 

593.  Let  it  be  required  to  develop  (a-\-xy,  for  any- 
rational  value  of  n,  into  a  series  of  the  descending  powers 
of  X, 


THE  BINOMIAL   THEOREM, 

(a  +  ^)«  =  ja(l  +  f)["=a«(l+f)" 

Put-  =  ^,  or  (1+  -y=(l  +  ^)« 

"Put  {\  +  zY  =  I  +  Az  Jr  B z"^  +  C z^  ■\-  Dz/^  + . . , , 

Since  z  may  have  any  finite  value,  put  z  —  r\  then, 
(1  +  r)»  =  1  +  uir  +  i?r2  +  Cr3  +  i)r*  + . . . . 

Subtract  (D)  from  (C), 
(1  +  zY  -  (1  +  r)»  =  ^  (2;  -  r)  +  ^  (2«  -  r2)  +  C  (2«  -  r3)  +  . . . . 

Put  ^  f  or  1  +  2;  and  R  f or  1  +  r ;  or  Z  —  i2  f  or  2  —  r ;  then, 
Z^  -  R^  =  A{z  -  r)  +  B{z'^  - r«)  +  C{z^  -  r^)  + . . . . 

Divide  hj  Z—  R  =  z  —  r, 
Z 


329 

(A) 

(B) 
(C) 

(D) 

(E) 

(F) 

(G) 


Z  —  R  \  z  —  r  J         \  z  —  r  J 

Let  lim.  2;  =  r,  then  lim.  Z=R,  since  Z=l  +  z,  and  i?  =  1  +  r ; 

andlun.(§=|-")z=B  = 

.-.  nR^-^  [592] 
or,  n(l  +  r)"-* 


(H) 


J.  +  2  J5r  +  3  02  +  4i)r3+ . . . . 


(1) 
(J) 


Multiply  by  1  +  r,  and  column  coefficients, 


w  (1  +  r)* 


+  2B 


r^  +  dC 

+  4:D 


r3  + 


(K) 


r-¥2B 
+  3(7 
Multiply  (D)  by  n, 

n(l  +  r)»  =  71  + J.nr +  -e7ir2  + Cwr»  +  Z^nr4+ (L) 

Equating  the  coefficients  of  the  second  members  in  (K)  and  (L), 
we  have 

(l)A  =  n  (2)  A  +  2B  =  An 

{^)2B  +  dC  =  Bn  (4)  3C  +  4i>  =  i>w;  etc. 


A  =  n,  B  =  — g— ,  C= ^ 


n{n 


D='''      ^''^      ^,etc. 

Substituting  the  values  in  (C), 

{1  +  zY  =  l  +  nz+  -^ — '-z^  +  r£ 2»  + 

n(n-l)(n-2)(n-S)^,_^^^^^        ^^^ 


330  ADVANCED  ALGEBRA. 

Substitute  —  for  z  (B),  and  multiply  by  o*»  (A), 

Cb 

n(7i-l)(w  — 2)a"-3  ,      w(7i  — l)(w  — 2)(n— 3)        ,   ^  '      ^t. 

—5^ '-j^ x'+— '^         '^ ^-a^-*x*+ (N) 


594.  Cor.  1.     {X  +  i/)"  =  a-  ^1  +  ^= 

=  X"  +  «a?-' y  +  ^i^^af-^/ + 

"('^-^>("-^>x-V+etc.        (P) 

This  is  the  most  general  form  of  the  binomial  theorem,  inasmuch 
as  X  and  y  may  be  both  variables. 

595.  Cor,  2, — By  inspection  it  will  le  seen  that 

1,  The  rth  term  of  the  development  of  {x  -\-  yY  = 
n(n-l){n-^)..,.{n-r-^%)    ^«_,+,   ,_i 

|r  —  1  *  ^ 

2,  The  {r  +  l)th  term  = 

n{n^l){n-^2).,.,{n-r  +  2)(n-r  +  l) 

|r-lxr  =  |r  '^      ^ 

3,  The  ratio  of  the  (r  -f-  l)th  term  to  the  rth  term  = 
n^r-\-l    y 

r 


x'      '  {     r  )  X 


596.  Cor,  3, — 1.  If  n  is  a  positive  integer  equal  to 
r  —  1,  the  coefficient  of  the  {r  +  l)th  term,  which  is  also 
the  coefficient  of  the  (n  +  2)th  term,  will  reduce  to  zero. 
Therefore,  the  series  will  terminate  with  the  {n-\-l)th 
term,  which  will  he  y\ 

2.  If  n  is  negative  or  fractional  no  factor  of  the  rth 


THE  BINOMIAL   THEOREM.  331 

term  (r  being  a  positive  integer)  will  reduce  to  zero,  how- 
ever great  r  he  taken.    Therefore,  the  series  will  he  infinite. 

597.  Cm-.  4,— 

Since  Urn,  \  I  —^ -'^ll'       =  —1  ,^ ,  it  follows : 

1.  That  the  coefficients  of  all  terms  in  the  hinomial 
theorem  are  finite  hoioever  far  the  theorem  he  expanded. 

2.  That  if  y  <x  the  expanded  form  is  convergent  [583]. 

3.  That  if  y>  X,  the  literal  part  {not  coefficient)  of  the 
{r  +  Vjth  term  will  increase  indefinitely  as  r  increases  and 
will  ultimately  become  infinitely  great,  and  as  the  coefficient 
remains  finite  the  whole  term  will  become  infinitely  great. 
Therefore  the  expanded  form  will  be  divergent  [580]. 

Jf..  If  y  ■=.  ±x  the  expansion  will  be  indeterminate  ; 
hut  (x  +  y)''  =  (2  a:)"  or  (0)"  =  2"  a;*  or  0. 

5.  The  expansions  of  {x-\-yY  and  {y-\-xY  can  not 
both  be  convergent  for  particular  values  of  x  and  y  ;  only 
the  one  that  has  the  greater  first  term. 

598.  Cor,  5, — 1.  The  coefficient  of  the  rth  term  will  evi- 

n  "^  r  I  1 
dently  be  greatest  when  ^^—  is  first  <  1 ;  or  when 

n  —  r-\-l  is  first  <r ;  or  when  2  r  is  first  >  w  +  1,  or 

when  r  is  first  >     T"    . 

2.   The  rth  term  when  the  expansion  is  convergent,  or 

n  ~~  r   I   1      1/ 

when  xy  y,  is  evidently  greatest  when  ^^^— -  .  —  is 

r  X 

first  <  1 ;  or  when  {n  —  r-\-\)y  is  first  <,rx ;  or  when 
{n-\-l)  y  is  first  <  {x-\-y)  r  ;  or  when  r  is  first  >  (  -^—  |  y. 

Illustration.  — In  the  expansion   of    (8  +  -^)      ,    the 


332  ADVANCED  ALGEBRA. 

greatest  coefficient  belongs  to  the  term  whose  number  is 

—  4  +  1         1 
first  greater  than  — ^ —  >  ^^  TKy  which  is  the  first  term. 

The  greatest  term  in  the  expansion  of  I  8  +  -^  j       is  the 

one  which  immediately  follows  in  number,  — \7~     x  -5- , 
1  .  .  ^t  ^ 

or  — ,  which  is  again  the  first  term. 
00 

EXERCISE    91. 

Expand  : 

1.  (a  -  3  T'Y  3.  (a;  -  3  af  5.  {x^  -  5)8 

2.  (2  +  5  xY  4.  (2a^-\-  5y  6.  (3  x^  +  a^y 
Expand  to  four  terms  : 

7.  (1  -  x)i  9.  (a^-  1)^  11.  («rr  +  b)-^ 

8.{a—x)^  10.  (a;^  +  «)~^  12.  {x^  —  a^~^ 

13.  Extract  the  cube  root  of  126  to  six  decimal  places. 
Suggestion : 

Vl26  =  Vi25Tr  =  (125  +  l)i  =  j53(l  +  ^)P  =  5(l  +  ji^)* 

-^r       3       125+       \2_       ""  V125;  +  [3_ 

(^y+  etc.  [  =  5(1  +  0026666  -  -0000071  +  -0000001)  =  5-0132975. 

14.  Find  to  5  decimal  places :  V65,  VSO,   V344,   V3128 

15.  Find  the  7th  term  of  {2x-}-SY^ 


16.  Find  the  5th  term  of  V4+^ 

a 

17.  Find  the  6th  term  of 


Va-\-x 

18.  Find  the  rth  term  of  {a  —  x)~^ 

19.  Find  the  greatest  coefficient  of  (2  +  x)^ 

20.  Find  the  coefficient  of  the  5th  term  of  {a  —  a^)~^. 

21.  Find  the  numerical  value  of  the  10th  term  of 

(7  —  5  3/^)",  when  y  =  27  and  n  =  S 


TEE  EXPONENTIAL   THEOREM.  333 

The  Exponential  Theorem. 

599.  The  exponential  theorem  is  the  expansion  of  a"  in 
ascending  powers  of  x,  and  is  derived  as  follows  : 

T>  ^  /^       lA""     -  1       nx(nx  —  l)    1 

And  (l  .  1)'=  1  .  1  +  ^  .  ^-— I^^^  etc.  (C) 
Substitute  (B)  and  (C)  in  (A), 

x{x-l)      x(x-^)(x-l) 

1  +  ^+ — \r^*- \r — -^'"'- 

Suppose  lim.  n  =  ao,  then 

Put  e  for  1  +  1  +  i-ft-  +  fo"  +  iT"  +  6tc. ;  then, 

[£_     l£_     l± 

e»  =  l+a:  +  j^  +  --  +  g-+  etc.  (F) 

Put  ca;  for  a;,  then  e""  =  1  +  ex  +  -rjr-  +  -r^-  +  etc.  (G) 

Let  e"  =  a,  and  assume  e  as  the  base  of  a  system  of  logarithms 
[465],  then  c  =  loge  a,  read  logarithm  a  to  the  base  e.  Substitute 
these  values  in  (G), 

»-  =  l  +  ^log.a+^<!2|^  +  ?!(l5|^  +  eto..  (H) 

which  is  convergent  for  all  finite  values  of  x  [582]. 
This  is  the  Exponential  Theorem. 

600.  SchoUum.  e  =  l  +  l+-i-  +  J-  +  ,-^  +  etc.  =  2-7182818. . . . 

l^      l£-      lL 
is  the  base  of  the  Napierian  or  natural  system  of  logarithms,  a  system 
universally  used  in  theoretical  work  instead  of  the  system  based  on  10, 
which  is  used  in  practical  work  only. 


334  ADVANCED  ALGEBRA. 

The  Logarithmic  Series. 

601.  The  logarithmic  series  is  the  expansion  of  loge 
(1  -f-  ^)  in  the  ascending  powers  of  x,  and  is  derived  as 
follows  : 

a>  =  l  +  ,log.  a+  t^^  +  t^^^  ,t,.  p99,H].  (A) 

Transpose  1  and  divide  by  y, 

-y-  =  logea  +  2/f-^  +  '-^"-^  +  etcj.  (B) 

Let  lim.  y  =  0,  then 

loge  a  =  lim.  I I 

Put  1  +  a;  for  a,  then 
loge  (1  +  a;)  =  lim.  -  ^1  +  xy -l\-^^^ 

y  r         12.  |3_  ij/=o 

=  a;  —  -^r^  +  -o  —  etc.    Therefore, 

lO  o 

logo  (l+a:)  =  aj-^a;2+  -^x^—  jX^  +  etc.  (C) 

This  is  known  as  the  Logarithmic  Series. 

602.  The  ratio  of  the  (w  +  l)th  term  to  the  nth.  term 
x'^'^'^     x""  ^      n  _      1 

IS  I     7  !  —  j    —  .  X  —  r-  .  X . 

n-{-l     n       n-\-l  i_4__ 

Now,  lim.  / .x\  =x.    Therefore,  if  a;  <  1,  numerically 


li+i'7„=» 


the  series  is  convergent.    It  is,  therefore,  convergent  for  all  values 
of  X  between  —  1  and  +  1. 

When  x  =  l,  loge  2  =  1—  —  +  -^  —  -j  +  etc.,  which  is  converg- 
ent [5831. 

When  x=  —  l,  log,  ^=-'^('^+2+'o+T  +  ^^^-j'  which  is 

divergent,  since  lim.  (  -^-^  )        =  lim.-!  —1  ( r  |  >■         =  —  1, 

and  all  the  terms  have  the  same  sign  [583]. 


THE  LOGARITHMIC  SERIES.  335 

603.  Eesume 

Put  —  X  for  oCf  then 
\og.{l-x)^-x-\x^-\7?-\^-\o?-....    (3) 

Subtract  (3)  from  (1),  then 
log.  (1  +  *)  -  log.  (1  -  X),  or  log,  ([±|)  [467,  P.  3]  = 

2(.+  J  +  |'+....)         (3) 

171  —  n  ^  ' 

Put  — I —  for  X,  then. 
m-{-n 

Put  m  =  n-{-l,  then 

iog,(„+i)-iog.«=2J^+i(^^y+ 
iog.(«+i)=iog.»+3J^-^+j(^^y+ 

+  ....[  (D) 


5  \2  ^  +  1/ 


As  this  formula  converges  very  rapidly  for  all  values 
of  n,  it  may  be  used  to  find  the  Napierian  logarithm  of 
any  number  from  that  of  the  next  preceding  number, 
n  being  regarded  an  integer. 


Computation  of  Logarithms. 

604.  The  logarithms  of  composite  numbers  may  be 
readily  found,  when  the  logarithms  of  primes  are  known, 
by  Art.  467,  P.  2. 


/ 


OF  THE 


336  ADVANCED  ALGEBRA. 

The  logarithms  of  prime  numbers  are  found  by  formula 
D,  Art.  603. 

niustrations. — 
loge    1  =  0  [466,  P.] 
loge    2  =  loge  (1  +  1)  = 

^  "*"  ^  \  3  +  d^iW  +  5l<^  +  7x~3^  +  •  •  •  •  j 

=  0-69314718 (by  actual  reduction). 

loge    3  =  loge  (2  +  1)  = 

=1-09861228.... 
loge    4  =  2  loge  2  =  1  -38629436 .... 

loge      5=   loge   (4  +  1)    = 

loge4  +  2(i  +  3^3  +  5^  +  ....) 

=  1-60943791.... 
loge    6  =  loge  3  +  loge  2  =  1  '79175946 .... 

loge      7   =  loge  (6  +  1)   = 

loge  6  +  2(1    +   3^  +   ^^  +  ....) 

=  1-94591.... 
loge    8  =  31oge2  =  2-07944 
loge    9  =  21oge3  =  2-19722 
loge  10  =  loge  5  +  loge  3  =  2-30258509 
etc.,  etc.,  etc. 


605.  Let  a  and  i  represent  the  bases  of  two  systems 
of  logarithms  and  n  any  number. 

Let   logb  n  •=  Xf  then  l'  =  7i  (1) 

Let   loga  l  —  m,  then  oT  =.1)  (2) 

dT"  z=:  If  =:  n,  or  log.  n  =  mx  (3) 

loff.  n      mx 
.*.  ,  ^       =  —  =  m :  or 
logb  n        X 

log,  n  =  m  logb  n.     Therefore, 


COMPUTATION  OF  LOGARITHMS.  337 

Principle. — Multiplying  the  log^  of  a  number  hy  the 
loQa  of  b  gives  the  log^  of  the  number. 


606.  Loge  n  =  logio  n  X  loge  10  [605,  P.] 

.-.  log.o^  =  loge^  X  j^  =  log.^  X  ^^30^ 

=  loge /J  X  0-4342944.... 

607.  The  number  0-4342944 is  called  the  modulus 

of  the  common  system,  and  is  represented  by  m. 

Therefore, 

JPrin,  2, — Log^Q  n  =  m  log,  n. 

By  means  of  this  principle  the  Briggean  or  common 
logarithms  may  be  derived  from  the  Napierian  or  natural 
logarithms. 


I.  Since  loge  (^  +  1)  = 

logio  (»  +  !)  = 
]og.„»t  +  3mj^^  +  i(^^y+....|      (H) 

By  means  of  this  formula  the  common  logarithms  may 
be  computed  directly. 


Summation  of  Series. 

609.  No  general  method  of  summing  series  can  be 
given.  Series  of  special  types  may  sometimes  be  summed 
by  special  methods.  The  student  has  already  learned  how- 
to  sum  an  arithmetical  progression  [486],  a  geometrical 
progression  [493],  an  infinite  series  of  the  geometrical 
type  [499],  an  arithmetico-geometrical  progression  [«500], 
a  series  of  square  numbers  [506],  a  series  of  cubic  num- 


338  ADVANCED  ALGEBRA. 

bers  [507],  and  series  dependent  upon  or  resolvable  into 
these.     A  few  additional  methods  will  be  given  here. 

610.       I.  Method  by  Indeterminate  Coefficients. 

This  method  is  applicable  when  the  nih.  term  is  a 
rational  integi-al  function  of  n. 

niustration. — Find  the  sum  Sn  of  n  terms  of  the  series : 
lX22  +  2x32  +  3x4:2  +  ...,  +  w(w  +  1)2. 

Solution : 

Put        Sn=   1  X22  +  2x32  +  3x42  +  4x52  +....+  7i(W+l)2 

==  ^ +  ^n  +  (7n2+i>w3+^7i4+ 

and,  Sn+\  =  1x22 +  2x32 +  3x42  + 4x52 +....+  (?i  +  l)(n  +  2)2 
=  ^  +  ^(71  +  1)  +  (7(?i  +  l)2 +Z>(n  +  l)3  +  ^(n  +  l)4+.... 
Then,  by  subtraction, 
(w+l)(w  +  2)2  =  ^  +  (2w  +  l)(7+(3n2  +  3n  +  l)i) 

+  (4n3  +  6n2  +  4?i  +  l)^  + ,  or  w3  +  5n2  +  8w  +  4  = 

{B+G+D+E)  +  {2G+^D  +  4.E)n  +  {^D+QE)'n?  +  4:En\ 
since  all  coefficients  after  E  are  zero,  there  being  no  more  than  four 
terms  in  the  expansion. 

Equating  the  coefficients  of  the  like  powers  of  ti, 
1.4^=1,  2.  3i>  +  6^=5, 

3.  2C+3i)  +  4^=8,  4.  5  +  C+Z)  +  ^  =  4; 

17  7  5 

whence,  E  =  -^,  7)  =  -^,   C  =  -r,  and  B  =  -^» 

5  7  7  1 

Sn=  A+  -^n+  -^'n?  +  -^n^  +  -jn^ 

To  find  A,  put  n  =  1,  then  Sn  =  Si  —  the  first  term  =  1  x  22. 

...(lx2)2  =  A+|  +  |  +  |-  +  i  =  ^  +  4; 

whence  A  =  0;  and 

5  7  7  1 

=  i^(3n*  +  14  n8  +  21^2  +  lOn) 
la 

=  ^(3n»+147i»+21»+10)=^(w+l)(n+2)(3w+5) 


SUMMATION  OF  SERIES.  339 

611.  2.  Method  by  Decomposition. 

This  method  is  sometimes  applicable  when  the  wth 
term  is  a  rational  fractional  function  of  n,  and  is  resolv- 
able into  the  algebraic  sum  of  the  nth.  terms  of  two  or 
more  other  series  of  the  same  nature. 

niustration. — 

Find  the  sum  S,,  of  n  terms  of  the  series :  ^r — ^ — 7  + 

7         ,        10        ,  ,  37^  +  l 


3x4x5    '    4X5X6    '  •**•  '    (/i  +  1)  (/^  +  2)  (/^  +  3)  * 

Solution : 

3/1  +  1  _     A  B  C 

^^^  (w  +  l)(w  +  2)(n  +  3)  -  n  +  1  "^  n  +  2  "^  71  +  3'  ^^®" 
A  =  —ly  B  =z  5,  and  C  =  —  4 ;  whence 

3n  +  l  _  / 1_         5 4_\ 

(n  +  l){n  +  2)(n  +  S)~  \     n  +  l'^n  +  2     n  +  dj' 

••     "  ](7i  +  i)(n  +  2)(w  +  3)f 

^V      n  +  lj  2       3       4 


\      n  +  2) 


4  n  +  1 

5       5  5 


■o    +^  + 


3      4 n  +  1     n  +  2 

2/ 4_\  _4_ 4 4 ^ 

V      ^  +  3/  4  71  +  1      n  +  2      n  +  S 

Adding  the  last  three  series,  we  have 

"~V      2"^3"^w4-2      n  +  s)~6'^n  +  2      n  +  3 

If  lim.  n  =  ao,  then  /Soo  =  77 . 
0 

The  Differential   Method. 

612.  If  the  first  term  of  any  series  be  taken  from  the 
second,  the  second  from  the  third,  the  third  from  the 
fourth,  and  so  on,  a  new  series  will  be  formed  which  is 


340  ADVANCED  ALGEBRA. 

called  the  first  order  of  differences.  If  the  first  order  of 
differences  be  treated  in  the  same  manner  as  the  original 
series,  a  second  order  of  differences  will  be  formed,  and 
so  on. 

Thus,  if  we  let  a,  I,  c,  d,  e,  , , . ,  be  any  series,  then 

h  —  a,  c  —  h,  d—  Cy  e  —  d,  ....  will  be  the  first  order 
of  differences ; 

c  —  %'b-\-a,  d  —  2c-}-b,  e  —  2d-^c,   the  second 

order  of  differences ; 

d  —  dc-\-3b-a,    e  —  Zd^^c-h,    the    third 

order  of  differences,  and  so  on. 

613.  If  we  let  «!,  Ji,  Ci,  fZi,   represent  the  first 

order  of  differences ; 

«2?  ^2?  Cgj  ^2*  •  •  •  •  the  second  order  of  differences ; 

^3 J  ^3>  ^3?  ^3^  ••••  the  third  order  of  differences; 
and  so  on,  we  have  the  following  scheme  : 
Series :  a,    l,    c,    dy    e, 

1st  Differences:     ^i,   Jj,   Ci,   (?i, 

2d    Differences :  ct^,   hi   c^y 

3d    Differences :  ^3?   ^3? 

4th  Differences  :  ^4,  ,  and  so  on. 

(hi  ci2f  (hy  <^if are  the^r^^  terms  of  the  succes- 
sive order  of  differences. 

614.  Problem  1.    To  find  the  {n  +  l)th  term  of  a  series. 

Solution :  Take  the  series  a,  6,  c,  d,  e, then  from  the  above 

scheme, 

1.  b  —a  =ai,  whence  b  =  a  +  ai  (1) 
bi  —  ai  =  a%,  "  bi  =  ai  +  a^  (2) 
&a  —  aa  =  Os ,  "  63  =  O9  +  as  (3) 
&3  —  as  =  a4 ,        "       ba  =  a3  +  at  (4) 

2.  c  =b  +bi  =  a  +2ai  +  tti,  from  (1  and  2)  (5) 
Ci  =  &i  +  6a  =  ai  +  2  as  +  as ,  from  (2  and  3)  (6) 
Ca  =  Ja  +  6s  =  as  +  2  as  +  a4 ,  from  (3  and  4)  (7) 


SUMMATION  OF  SERIES.  341 

3.  <?  =  c   +  Ci  —  a   4-  3  «!  +  3  aa  +  aa ,  from  (5  and  6)  (8) 
<?i  =  Ci  +  Ca  =  «!  +  3  aa  +  3  aa  +  a  4,  from  (6  and  7)              (9) 

4.  e  =  6^  +  cZi  =  a  +  4  «!  +  6  CTa  +  4  as  +  ^4 ,  from  (8  and  9)     (10) 
Now,  since 

&r=a  +  ai,  c  =  a  +  2ai  +  aa, 

(?  =  a  +  3aj  +  3aa  +  «3,  e  =  a  +  4ai  +  6a3  +  4a8  +  a4, 

it  will  be  observed  that  the  coefficients  of  any  term  are  the  same  as 
the  coefficients  of  a  power  of  a  binomial,  whose  index  is  one  less  than 
the  number  of  the  term.    Hence,  the 

/      H•.l^  X                              n(n—\)          w(w— l)(w— 2)  ^., 

(71+ l)th  term  =  a+w  Oi  +  -^ — -  a^  +  — j^^ ~  aa  + [A] 

615.  Car, — The  nth  term  = 

;+in-l)a,+  ^''-^^^-'K  +  ....       [B] 

Illustrative  Example. — Find  the  7th  term,  also  the  nth 

term,  of  the  series :  1,  3,  6,  10, 

Solution :  1st  differences  =  2,  3,  4,  . . . . 
2d  "  =    1,  1,     . . . . 

3d  "  =0,       .... 

.*.  1st.  n  =  7,  a  =  1,  ai  =  2,  a<,  =  l,  as  =  0 
Substitute  these  values  in  [B], 

7th  term  =  1  +  6x2  +  ^-^  x  1  =  28 

2d.  Put  n  =  n,  a  =  l,  ai  =  2,  aa  =  1,  and  aa  =  0, 

i-i,         i-u  i  -.      /        ^^      n      (n  —  l)(n  —  2)      .       7i(n—l) 

then  nth  term  =  l  +  (w  —  l)x2+  ^^ '-^ x  1  =  —^ — - . 

«  2 

616.      Problem  2.    To  find  the  sum  of  n  terms  of  a  series. 

Solution:  Let  it  be  required  to  find  the  sum  of  n  terms  of  the 
series  a,  b,  c,  d,  e, 

Assume  the  series  0,  a,  a  +  b,  a  +  b  +  c,  a  +  b  +  c  +  d, ,  then, 

1st.  The  first  order  of  differences  of  the  assumed  series  is  the 
given  series. 

2d.  The  second,  third,  and  ?ith  orders  of  differences  of  the  as- 
sumed series  are  the  first,  second,  and  {n  —  l)th  orders  of  differences 
of  the  given  series. 

3d.  The  {n  +  l)th  term  of  the  assumed  series  is  the  sum  of  n 
terms  of  the  given  series. 

Hence,  if  in  formula  [A]  we  put  0  for  a,  a  for  ai,  ai  for  aa, 


342  ADVANCED  ALGEBRA. 

etc.,  we  shall  have  the  sum  of  n  terms  of  the  given  series.    Doing  so, 

we  shall  have 

c,  n{n  —  l)  w  (n  —  1)  (n  —  2) 

Sn  =  na+     "^         'ax+  -^ ^ U^+..,,  [C] 

Note. — This  method  is  applicable  only  when  some  finite  order  of 
differences  will  reduce  to  zero. 

Example. — Find  the  sum  of  10  terms  of  the  series  : 
1  +  4  +  10  +  20  +  35+.... 

Solution :  First      Differences  =  3,  6,  10,  15, 

Second  Differences  =     3,   4,   5,       

Third    Differences  =        1,    1,  

Fourth  Differences  =  0,  .... 

.-.    Put    w  =  10,  a  =  1,  tti  =  3,  a,  =  3,  a^  —  1,    and  a^  =  0    in 
formula  [CJ ;  then, 

^       _^10x9       „10x9x8      _       10x9x8x7      , 
/S;  =  10  +  —5—  X  3  +      ^    .,      X  3  +  — s-^ — 7—  X  1  =  715 
3  2x3  2x3x4 


617.    Problem  3.  To  interpolate  terms  at  regular  inter- 
vals between  the  terms  of  a  given  series. 

Formula  [B]  may  sometimes  be  used  with  advantage 
to  interpolate  terms  at  regular  intervals  between  the  terms 
of  a  given  series. 

Illustrations. — 1.  Given 

(651)3  =  423801,  (653)3  =  426409, 

(655)3  =  429025,    and   (657)3  =  431645, 
to  find  the  value  of  (652)3,  (554)3^  ^^^  {'o^^f. 

Solution:         Series  =  423801,  426409,  429025,  431649 
First     Differences  =  2608,  2616,  2624 

Second  Differences  =  8,  8 

Third    Differences  =  0 

Take  formula  a„  =  a  +  (n  —  1)  «t  +  10  «s  +  . . . . 

L~_ 

Put  a  =  423801,  a,  =  2608,  a,  =  8,  and  7i=  l-^- ,  2  g,  and  3^ 
successively;  then, 


1.  (652)8  _  423801  +  -|-  x  2608  -  |  x  8  =  425104 

2.  (654)8  =  423801  +  |  x  2608  +  |  x  8  =  427716 

3.  (656)8  =  423801  +  |  x  2608  +  ^  x  8  =  430336 


SUMMATION  OF  SERIES.  343 

V65T=  8-666831,     V652'=  8-671266, 
V653  =  8-675697,     V654  =  8-680123,  and 


V655  =  8-684545,  find  V653-75. 

Solution :    Here   a  =  8-666831,    «i  =  -004485,    a,  =  -  0-000004, 

tta  =  —  0-000001,  0^4  =  0,  and  tj-  =  3  -j .    Substitute  these  values  in 

n        n^(.  in-\){n-^)  (n-l)(7i-2)(n-3) 

a«  =  a  +  (71  —  1) «!  H r^ tta  H rK ^s ;  then, 

li  /T'J'  OQ1 

V653-75  =  8-666831  +  ^  x  '004435-  ^  x  -000004-  g^^  x  -000001 
=  8-666831  +  -012196  -  -000009  -  -000001  =  8-679017. 


EXERCISE    92. 

Find  the  wth  term  and  the  sum  of  n  terms  of  the 
following  series  : 

1.  2  +  6  +  12  +  20  +  ....        4.  3  +  8  +  15  +  24  +  .... 

2.  1  +  9  +  25  +  49  +  ....         5.  1  +  4  +  9  +  16  +  .... 

3.  1  +  3  +  6  +  10  + . . . .  6.  2  +  12  +  30  +  56  + . . . . 

7.  6  +  24  +  60  +  120  +  210  +  .... 

8.  45  +  120  +  231  +  384  +  585  +  .... 

9.  1.3  .22  +  2.4.32  +  3  .5.42  +  .... 
10.  3. 5. 7  +  5. 7. 9  +  7. 9.  11  +  .... 

Sum  to  n  terms  and  to  infinity  : 
1.1.        1 


344  ADVANCED  ALGEBRA. 

Find  the  value  of  : 
16.  %{n^{^n-2)}  17.  :S  j|(w  +  l)(^  +  2) 

20.  The  log.  950  =  2-977724,     log.  951  =  2-978181, 
log.  952  =  2-978637,     log.  953  =  2-979093, 
find  log.  952-375. 


Reversion  of  Series. 

618.  If  2/  is  a  serial  function  of  x,  then  may  x  be  de- 
veloped into  a  serial  function  of  y,  and  the  process  is 
called  Reversion  of  Series. 

Example. — Kevert  y  =  a-\-lx-\-cx^+da?-{-ea^-\-,,,. 
into  a  serial  function  oi  y. 

Solution :  Put  the  series  in  the  form 

y—a  =  l)x-{-cx^-{-da^-\-ea^ 

Vvitx=zAiy-a)^B(jj-af  +  C{y-af  +  D{y-ay-\-.... 
Now,  hx  =bA(i/-a)  +  bB(i/-af  +  bC(y-af+bD{y-ay+.... 

cx^  =  cA^  (y-a)^  +  2  c  A  B  (y-af  +  {c  B^  +  2c  AC)(y-ay+ 

d a^  =  d  A^ (y-af  +  3d  A^ B {y-a)*+ . . . . 

ere*  =  eA*{y—ay  + 

,\  y-a  =  bA(y-a)  +  (bB+cA^)(y-af 

+  {b  C+2c A B+d A^)(y-af 
+  {b D  +  cB^  +  2c  A  C+2d  A^ B+eA*)(y-a)*+ . . . . 
Equating  the  coefficients, 

1.  bA  =  l;  whence  A  =  -j- 

2.  bB  +  cA^  =  0;  whence  B=-  ^ 

3.  bC+2cAB  +  dA'  =  0;  whence  C=  ^^^^^-^ 

4.  bD  +  c&  +  2cAC  +  SdA^B-\-eA*  =  0; 

¥e  —  5bcd  +  5c^ 
whence  D  = jji • 

/.    x  =  j{y-a)-y^{y-af+ ^ (y-af 

bU  —  5bcd  +  5c*, 
^i (y-a)*+.... 


REVERSION  OF  SERIES.  345 

619.  Cor.— It  a  =  0,  then 
1  c    3  ,    2c^-bd   ,      b^e-5bcd+6€^   .  , 


EXERCISE    93. 

Revert  the  following  serial  functions  of  x  into  serial 
functions  of  y: 
1.  y=zx-\-x^-\-a^-\-xf^-\- 

3.  y  =  x-]-2x^-{-63^  +  Ua^-\-.,J, 

4.  y  =  a;  +  a:^  +  2a;^  +  5a;^  + 

Suggestion.— Let  x  =  Ay  +  By^  +  Cy^  +  Dy''  + 

5.  y  =  l+a;+i^2+ia^+^a;*  +  .... 

6.  y  =  l  +  a;-2a:2_j_^ 

7.  Find  one  value  of  x  in  the  equation  a^-{-43^ -]-6xz=l 
Suggestion. — 

Put  1  =  2/,  and  assume  x  =  Ay  +  By^  +  Cy^  +  Dy*+..., 

8.  Find  one  valut  of  a;  in  ic^  +  l^^  x  =  l 


Recurring  Series. 

620.  A  series  in  the  ascending  powers  of  x,  in  which 
each  term,  after  one  or  more  fixed  terms,  is  px  times  the 
preceding  term,  or  px  times  the  preceding  -\-qa^  times 
the  next  preceding  term,  or  ^  a;  times  the  preceding  -{-qx^ 
times  the  next  preceding  -]-rx^  times  the  next  preceding 
term,  or  and  so  on,  is  a  Recurring  Series, 

621.  A  recurring  series  is  of  the  firsts  second^  or  nth 
order,  accordingly  as  each  term,  after  the  law  begins,  is 
derived  from  one,  two,  or  n  preceding  terms. 


346  ADVANCED  ALGEBRA. 

622.  The  forms  px,  px-{-qx^,  px-\- qx^  -\- rx^,  and 
so  on,  are  called  the  order  scales  of  the  series ;  and  p, 
p-\-Qy  i?  +  5'  +  ^^  and  so  on,  the  order  scales  of  the  co- 
efficients. 

Illustrations. — If  we  put  p  =  3,  q  =  4:,  and  r=  —2, 
then 

1.  2-{-(ix-\~lSx^-\-64:X^  +  ....  isaseriesof  the  first 
order. 

2.  2-\-6x-^26x^-{-102x^-\- is  a   series    of    the 

second  order. 

3.  2  +  6a;  +  26a;2  +  98rz;-^  +  386i5^  +  ....  isaseriesof 
the  third  order. 

623.  Prohlem  1.   To  determine  the  scale  of  coeflacients. 

1.  Let  a-{-I)x-\-cx^-\-da^-\-...,  be  a  recurring  series 
of  the  first  order. 

Then,  bx  =  apx;  cx^=p  bx^  ;  dx^  =pcx^ ;  and  so  on. 

b  c  d         ^ 

i^  =  -  ;         ^  =  ^  5  i?  =  -  ;  and  so  on. 

2.  Let  a-\-hx-\-cx^-\-dx?-\- be  a  series  of  the 

second  order. 

Then,  a q x^ -{-h p x^  =  cx^  (1);    l)qx^-\-cpoi^  =  da?  (2); 
whence,        aq-\-lp      =c      {^)',    hq     -\-cp     =:  d      (4). 
Then,  by  elimination, 

_  be  —  ad       ,       _bd—(? 
^  ""   b^  —  ac  ^  ~    ¥  —  ac 

3.  Let  a-{-bx-\-cx^-\-da?-\-ex''-\-fx^-{-....  be  a 
recurring  series  of  the  third  order.     Then, 

1.  ar-\-bq-{-cp  =  d  2.  br-}-cq-\-dp  =  e 

3.  cr-\-dq-\-ep  =zf 

By  elimination  the  values  of  p,  q  and  r  may  be  found. 
In  the  same  manner  the  scale  of  coefficients  of  a  recurring 
series  of  any  order  may  be  found. 


RECURRING  SERIES,  347 

624.  SchoHum, — In  order  that  the  scale  of  coefficients 
of  any  recurring  series  may  he  found,  there  must  le  given 
at  least  twice  as  many  terms  of  the  series  as  there  are 
terms  in  the  scale.  In  the  exercises  concluding  this  sub- 
ject just  twice  as  many  terms  of  each  series  will  he  given 
as  are  contained  in  the  scale  of  the  series.  This  will 
enable  the  student  to  determine  at  a  glance  the  order  of 
the  series. 

When  this  law  of  notation  is  not  followed,  as  when  the 
nth  term  of  a  series  only  is  given,  it  is  usually  hest  to 
expand  the  series  and  malce  trial  for  a  scale  of  two  terms, 
and,  if  the  results  thus  obtained  will  not  satisfy  the  series, 
then  make  trial  for  a  scale  of  three  terms,  and  so  on  until 
the  proper  scale  is  determined. 

Example. — Find  the  scale  of  coefficients  in  the  series 
l-{-%x-\-^x^-\-^o(?-\' and  expand  the  series. 

Solution :  This  is  a  series  of  the  second  order,  since  four  terms  are 
given.    Hence, 

1.  g  +  2i?  =  3  2.  2  g  +  3i?  =  4 

whence,  p  =  2  and  q  =  —1,  and  the  scale  is  2  —  1.    The  series  is 
1  +  2a:  +  3a;2  +  4a;3  +  5a;*  + nos^-h 

625.  Problem  2.    To  find  the  sum  of  n  terms  of  a 

recurring  series. 

The  method  of  finding  the  sum  of  a  recurring  series 
is  the  same,  whatever  be  the  scale  of  the  series.  For  the 
sake  of  simplicity  we  will  here  assume  a  series  of  the 
second  order,  whose  scale  is  p-{-q,  for  illustration. 

Let  «o  +  «i  ^  +  «^2  ^  +  •  •  •  •  «»-i  ^^^  be  a  series  of  the 
second  order.     Then, 

>S„  =  «o  +    aiX-{-    a^x^  -\- +    «„_!  x'"'^ 

—pxSn=      —pa^x—pa^T?  — — ^«„_2a;"~* 

—  qx^S^  =  —qa^x^  — —  qa^^^af-^ 

—  q  a^-z^""  —  q  cin-\^'^'^ 


348  ADVANCED  ALGEBRA. 

Adding  and  remembering  that  the  sum  of  the   co- 
efficients of  each  term  from   the  third  to  the  nth.  in- 
clusive is  zero, 
(1  —px  —  q 0^)  /S'«  =  «o  +  («i  — i? «o) ic 

-  (P  «n_i  +  q  a^-z)  x^-q  a^-i  a;"+i ; 

whence,  S.  =  ^o  +  {a.-pa,)x 

1  —px  —  qar 

{pan-i-{-qan-2)x''-{-qan-i3f^'^ 
1  —px  —  qx^ 

626.  Cor, — If  X  <1  and  Urn.  n  =  co,  then 
^    ___  ao-\-{ai—pao)x 
1  ^px  —  qoir 

Example.— Sum  l^^x^bx^-\-l^ a;3+48 rr^+145 x^-{- 
....  to  7  terms  and  to  infinity,  when  x  <1, 

Solution:  1.  5jt?  +  Sg-  +  r  =  18  3.  18^9  +  5g  +  3r  =  48 

3.  48p  +  18^  +  5r  =  145 
.*.    p  =  2,  g  =  3,  and  r  =  —  1. 
S-,  =  l  +  3a:+5a;2  +  18a;8  +  48a;4+1453:5^.4i6a;6 
-2  a;  /S't  =    -2a;-6a;2-10a;8-86a;4-  96a;«-290a;«-832a;'' 
-3a;2,S'T=  -3a;2-  9a;8-15a;4_  54a;*-144a;«-435a;''-1248ic8 

+    x3>S'7=  +     x^+  3a;4+     5^^+  iga^e^  48a;7+  145a:« 

+  416a;9 
.-.    (l-2a;-3a;2  +  a;3)>S'7  =  l+a;-4a;2-1219a;''-1103a;8  +  416a:» 
l  +  a;_4a:2_1219a;'-1103a:8  +  416a;9 


/St 


1+X  —  4:X^ 


l-2a;-3a;2+a;» 

627.  If  /S'ao  is  developed  into  a  series  by  the  method 
of  indeterminate  coefficients,  the  original  series  may  be 
reproduced  to  any  number  of  terms  desired.  Therefore, 
/Soo  is  often  called  the  generatrix  of  the  series. 

628.  If  the  generatrix  can  be  decomposed  into  partial 
fractions,  the  general,  or  ni\i  term,  of  the  series  may 
easily  be  obtained,  and  hence,  too,  the  sum  of  n  terms. 


RECURRING  SERIES.  349 

niustration. — Let  it  be  required  to  find  the  nth  term 
and  the  sum  of  n  terms  of  the  series 

Solution :  It  may  readily  be  determined  that  i?  =  2,  g  =  3,  and  the 
l  +  3a;  11  3  1 

generatrix  =  ^_^^_.^^,  =  "  2  '  iT^  +  3  '  T^Ts^' 

11  1.11, 

^^^'-2  •n:T-=-2  +  2^-■2^+•••• 
r  _  l^n    ^  _  (-1)"^"-^  r493  b1  • 

3  1  3        9         27  , 

^^^        2T335=2  +  2^+-2^  +•••• 

_  (-l)na;n_l  3n+la;n_3 

'^''~       2a; +  2       "^      6a;- 2     ' 

2^—1  3»i 

nth  term  =  (—  1)**  .  -g—  +  -g-  .  a?*-*. 


EXERCISE    94. 

Sum  to  infinity  : 
1.  l  +  3ic  +  5a;2  +  13a;3  +  .... 

3.  2  +  2a;  +  4a;2  +  14a;34-.... 

4.  3  +  2a;-7a;2- 38:^3-.... 

5.  l  +  2a;  +  3a;2+ll:r3^35^_j_121a;5  +  .... 

6.  l-3:zj  +  5a;2_^5^^13^_|.61^5_|.__ 

7.  Sum  l  +  2ir  +  9a;2  +  33iz^+ to  6  terms. 

8.  Sum  1  — 2a;— 7a;2  — 8a;3  4- to  7  terms. 

9.  Sum  2  —  a;  +  6a;2  —  14a;3  + to  8  terms. 

10.  Sum  l  —  ^x-{-^x^^Qo(?^x^  —  ^x^-\- to  9  terms. 

Find  the  nth.  term  and  the  sum  of  n  terms  of 

11.  l  +  2a;-82;2  +  20a;3- 

12.  1  +  5  a;  -f  9  a;2  4- 13  a;3  _^ 


350  ADVANCED  ALGEBRA. 

Decomposition  of   Rational  Fractional   Functions. 

629.  To  decompose  a  rational  fraction  is  to  find  two 
or  more  other  fractions  whose  sum  equals  the  rational 
fraction. 

630.  It  will  be  only  necessary  to  show  how  to  decom- 
pose proper  fractions,  as  all  improper  fractions  may  be 
reduced  to  mixed  numbers,  which  process  will  already 
lead  to  a  partial  decomposition. 

Principles. 

631.  1.  Any  rational  fraction  of  the  form  of 

p 

,     ■     .,   _.-,. j—j- — r  may  he  decomposed  so  that 

[X -f-  a)  \x -\-  0)  ....  {x-\-n) 


(x-]-a){x-\-I)) (x-{-n)       x-\-a      x-\-b  x-{-n 

Illustration. — Put 

3^24. 14:^-29       _     A       .      B      .       C  .,. 


{x-l){x-\-%)(x-^)       x-1    '   x-{-2   '   x-3 

Clear  of  fractions  and  arrange  the  terms  according  to  the  descend- 
ing powers  of  a;, 

3x^  +  Ux -29  =  (A  +  B  +  C)x^ -  {A  +  4:B -  C)x - 

{6A-SB  +  2C). 
Equating  the  coefficients  [566],  we  have 

(1)  A  +B+C=d  (2)  A  +  4B-C=  -U 

(S)  6 A-BB  +  2C=  29 
Finding  the  values  of  A,  B,  and  C  by  elimination,  and  substi- 
tuting them  in  (A),  we  obtain 

dx'^  +  Ux-29      _      2 d_  4 

{x  —  l)(x  +  2){x-S)~  x-l       x  +  2'^  x-S' 

632.  In  a  similar  manner  it  may  be  shown  that 


(a X -{- b)  (c X -\- d) {mx-\-n) 


ax-\-l)       cx-\-d      '  mx-\-n 


RATIONAL  FRACTIONAL  FUNCTIONS.  351 

633.  2,  Any  rational  fraction  of  the  form  of 

P 

{p(^  -{- a  X -{- h)  {x^  -\- c  X -\- d) {a^ -\- m  x -\- n) 

may  he  decomposed  so  that 

P 

{x"  ^  ax  ^h)  {7^  -\-  ex  ^  d)  . . . .  {x^  -\-  mx  ^  n)  ^ 

Ax  +  B  Cx-\-D  Mx-^JV 

~r  ^2  I   ^  ^  I   ,7  "T  •  •  •  •  ~r 


x^-\-ax-{-bx^-\-cx-\-d  x^-\-mx-{-n' 

„,    ,    ^.         T,  i.         4:X^-8x^-63^-Ux-{-3 
mustration.-Put  (^^^^i)(^_^^i)(^.^^^^) 

Ax  +  B  Ox-^D  Bx-\-F  .. 


^  x^-\-x-\-l    '   x^  —  x-\-l    '    x^-\-x^% 

Clear  of  fractions  and  arrange  the  terms  according  to  the  descend- 
ing powers  of  ic, 

4a;4  -  8a:3  -  5a;2  -  15a;  +  3  = 

{A  +  C  +  E)0!^  +  {B  +  2  C  +  D  +  F)a^ 

+  {2A-\-4.C  +  2D-[-E)x'-^{-A  +  2B  +  ^C+4:D  +  F)x^ 

+  (2A-^  +  2(7+3i)  +  ^)a;  +  (25  +  2i>  +  i^)  (1) 

Equating  the  coefficients  [566], 

(1)  A  +  C+E  =  0 

(2)  ^  +  2C+Z)  +  ^=4 

(3)  2A  +  4(7+2/>  + J^=  -8 
(4)^-25-3(7-4i9-i^=5 
(5)2A-^  +  2(7  +  3i;  +  ^=-15 
(6)  2^  +  2Z)  +  ^=3 

Finding  by  elimination  the  values  of  J.,  B,  C,  D,  E,  and  F^  and 
substituting  them  in  (A),  we  have 

4a4_8a^_5a;2_i5a;4.3         _ 
(a;2  +  ic  +  1) {x^-x+  1) (ic*  +  a;  +  2)  ~ 
3  4  5 


x^  +  X  +  1       x^  —  X  +  1       x^  +  X  +  2' 

634.  In  a  similar  manner  it  may  be  shown  that 

P 

{a  x^ -^  b  X -\- c)  {dx^-\-ex-\-f) {m  x^  -\- n  x -{■  p) 

Ax-\-B  Cx-\-D  Mx-\-N 


ax^-\-bx-{-c       d:x^  -\-ex  +/  mx^  -\-nx  -{-p  * 


352  ADVANCED  ALGEBRA, 

P 
635.   S,  A  rational  fraction  of  the  form  of  -, — ; — r- 
-'  J  J  J    {x-\-aY 

may  he  decomposed  so  that 

p  A  B  jsr 

t    /^   I    ^\2  ~r  •  •  •  •  "r 


(x-^aY       x-\-a       (x-\-aY  {x-\-ay 


Illustration. — Put 


i^-^r 


_     A  B  G  D 

"  x-%'^  {x-%)^^  {x-^f'^  {x-  2)*  ^^^ 

Clear  of  fractions  and  arrange  the  terms  according  to  the  descend- 
ing powers  of  x ;  then, 

3a:3  _  i4a;2  +  19a.  _  5  _  ^^3  _  (6  ^  _  ^)a;2  + 
(12J-4^+  C)x-{%A-4:B  +  ^G-D). 

Equating  the  coefficients,  we  have 

1.  ^  =  3  2.  6.4-5  =  14 

3.  12A-4:B  +  C=  19  4.  8^ -45  +  2  C- 2)  =  5 

Solving  these  equations,  and  substituting  the  values  of  A,  B,  C, 
and  D  in  (A),  we  obtain 

Sa^-Ux^  +  19x-5  3  4  1  1 


{x-2)*  x-2       (x-  2)2       (x  -  2f      {X  —  2)4 

636.  In  a  similar  manner  it  may  be  shown  that 

1  _p _^_+    -g    +    +    ^ 


(ax-i-by       ax  +  b   '    (ax-^-bf   '  '   (ax-^by 

2  P  ^     Ax-]-B 

(x^  +  ax-\-by       a^-\-ax-\-b^ 

Cx  +  D  Mx-^-N 


{^j^ax^bf   '  ••••  •   {x'-^-ax-irby 


{aT^^bx-^cy       a:^  +  bx-^c'^ 


Cx^D  Mx-{-JSr 


{as^  +  bx^cf   '  •••*  '    {a3^-\-bx-\-cy 

4.  Any  rational  fraction  whose  denominator  may  be 
resolved  into  linear  and  quadratic  factors  may  be  decom- 
posed by  a  combination  of  the  above  methods. 


RATIONAL  FRACTIONAL  FUNCTIONS.  353 

Ulustration. — 
To  decompose  ^^^_ a)^^-iY i^^+p^  +  ^y'  P^* 


{x  —  a)  {x  —  iy  {x^ -\- p  X -\- qY      x  —  a       x  —  b 

,        M  Px^Q  P'x-\-Q' 


{x  —  df       af-\-px-{-q  {x^ -\- p  x -{- q)* 

EXERCISE    93. 

Decompose  into  partial  fractions  : 

5cc  +  2  2a;-5  _      A  B 

^'     A   ^2  1  O  ^     I      -I        I 


x(x-{-l){x-\-2)        X    '   ic  +  1    '   x-^2 

3x-2  x^-x-{-l  ^    3x^-Ux-{-25 


(a;4-lf  (^  +  3)3  (a;-3)  (a;2-a:+6) 

aJ  +  (a  —  ^)a;  — ic^  ■      {px-{-qy 

^     ^  10. 


X^-{-X^  +  l  X(1-4:X^) 

3a:3_8a:2  +  10  2a;  +  l 

11<  7  TTT  12* 


(a;  -  1)*  (a;  -  1)  (a;^  +  1) 

13.   -n r^  14. 


a^^6x-{-6  (x-l)(x^-i-  ly 

X5.   .4t^+l^  16.  ^ 


(a;  +  1)  (x^  +  1)  -"•  a^  _|.  :r7  _  2^  _  ^ 
2a;g~lla;  +  5  5  ^^3  _p  g  ^  _j_  5  ^.^ 

(^~3)(a;2_|.2a;-5)  '  (ic^  _  j)  (^  _|_  1)3 

1  x^-{-x-\-l 

10     on    ■ 

^^-  a^^x'-x^-a^  (a;+l)(^'  +  l) 

21.       ^+^t^  .  22.  ^'  +  ^ 


l_a;-a.'*-|-a:«  "'••  {x-\-lf{x-2){x-{-3) 


CHAPTER  X. 
COMPLEX    J{ UMBERS, 


Graphical    Treatment. 

637.  If  a  straight  line  of  any  assumed  length  be  taken 
to  represent  the  number  one,  then  will  a  straight  line 
twice  as  long  represent  the  number  two,  one  three  times 
as  long  the  number  three,  and  so  on.  Thus,  we  see  that 
any  number  may  be  represented  by  a  line. 

638.  A  line  representing  a  number  is  called  a  graph 
number,  or  a  vector.  The  point  where  a  vector  is  sup- 
posed to  begin  is  called  the  origin,  and  the  point  where 
it  ends,  the  extremity. 

639.  A  vector  is  fully  determined  when  both  its  length 
and  direction  are  given.  In  a  system  of  graphical  repre- 
sentation of  numbers,  a  vector  running  rightward  from 
its  origin  represents  a  positive  number  and  is  positive,  and 
one  running  leftward  from  its  origin  represents  a  nega- 
tive number  and  is  negative, 

640.  If  the  vector  +«  be  made  to  revolve  about  its 
origin  A,  through  an  angle  of  180°,  or  ir,  it  will  become 
the  vector    —  a,   or  will  be 

multiplied  by    —  1 ;    and  if      j^,       ~  ^       i        "'" "        r^ 
the  vector   —  a  be  revolved 

about  its  origin  A  through  an  angle  of  180°,  or  tt,  it  will 
become  the  vector  -|-  a,  or  will  be  multiplied  by  —  1. 


GRAPHICAL   TREATMENT.  355 

Therefore, 

1.  Revolving  a  vector  through  an  angle  of  180°,  or  tt,  is 
equivalent  to  multiplying  it  hy  —  1. 

2.  Revolving  a  vector  about  its  origin  through  an  angle 
of  360°,  or  2  tt,  is  equivalent  to  multiplying  it  twice  hy 

—  1,  or  once  ly  + 1^  which  does  not  affect  its  length  or 
direction. 

3.  Since  —  1  =  V—  1  X  V—  1,  revolving  a  vector 
about  its  origin  through  an  angle  of  90°,  or  \  tt,  is  equiva- 
lent to  multiplying  it  by  V—  1,  or  i  [v.  P.  I.     299]. 

641.  Motion  about  the  origin  of  a  vector  in  the  direc- 
tion the  hands  of  a  clock  go  is  considered  negative,  and 
counter-motion  positive.  The  factor  -\-i  may,  therefore, 
be  taken  to  represent  circular  motion  about  an  origin 
through  an  angle  of  90°,  or  \  it,  counter-clock-wise ;  and 

—  i  through  an  angle  of  90°,  or  \  tt,  cloclc-wise. 

642.  Since  (+  i)  X  (+  i)  X  (+  i),  or  (+  if  =  -  i,  the 
factor  symbol  —  i  may  also  denote  circular  motion  about 
an  origin  through  an  angle  of  270°,  or  |  tt,  in  a  positive 
direction. 

643.  Since  {±  i)  X  (±  i),  or  (±  if  =  -  1,  the  factor 
symbol  —  1  may  denote 
circular  motion  about  an 
origin  through  an  angle  of 
180°,  or  TT,  in  either  direc- 
tion. 

Illustrations.  —  1.    The    B- ^ 

vector  -\-a  multiplied  by 
-{-i  =  AB  revolved  about 
A  in  the  positive  direction 
through  an  angle  of  90°  = 
AB\ 


B' 


+a 


—at 


356  ADVANCED  ALGEBRA. 

2.  The  vector  +  a  multiplied  by  —l^AB  revolved 
about  A  in  the  positive  or  the  negative  direction  through 
an  angle  of  180°  =  A  B", 

3.  The  vector  +  a  multiplied  by  —i  =  AB  revolved 
about  A  in  the  negative  direction  through  an  angle  of  90°, 
or  in  the  positive  direction  through  an  angle  of  270°  = 
A  B'". 

4.  In  a  similar  manner  it  may  be  shown  that  (—  a)  X 
(+^)  =AB"'',  (-«)x(-l)  =AB,  and  {- a)  X  {- i) 
=  AB\ 

644.  From  what  has  been  explained  thus  far  it  will  be 
seen  that,  if  a  vector  a  units  long  running  rigJitward  from 
its  origin  represents  +  «^  running  leftward  from  its  origin, 
it  will  represent  —  a ;  running  upward  from  its  origin, 
-\-ai'y  and  running  doiunward,  —  ai. 

646.  One  vector  is  added  to  another  by  placing  its  ori- 
gin to  the  extremity  of  the  other  and  giving  it  the  direc- 
tion  indicated  by  its  factor  symbal.  The  vector  of  their 
sum  is  the  length  and  direction  of  the  line  joining  the 
origin  of  the  vector  to  which  addition  is  made  with  the 
extremity  of  the  vector 
added.  ^         +«        ?       +5        ^ 


Illustrations. — 

1.  The  vector  B  C  ^ ±^ 

(=+&)  added  to  the  G 

vector       A  B  {=  -\-a)  —h 


■^^r-B 


gives    the  vector  A  C     ^  A       ~^^ 

(=  +  («  + 5)). 

2.  The  vector  B  C  (=  -  b)  added  to  the  vector  A  B 
(=  -f  a)  gives  the  vector  A  C  {= -{- {a  —  h)] ,  when  a>h. 

3.  The  vector  BO (=  —  !))  added  to  the  vector  AB 
(=  -f  a)  gives  the  vector  A  C {=  —  (b  —  a)},  when  a<b. 

Note.— To  subtracit  a  vector  is  to  add  the  vector  with  contrary  sign. 


COMPLEX  NUMBERS. 


357 


Representation  of  Complex  Numbers. 

646.  Let  it  be  required  to   represent  graphically  the 
complex  numbers  a-^hi,  a  — hi,  ^a-\-hi,  and  —a  —  hL 


1.  a^bi  =  +a+(+Ji). 

.-.  Vector  {a-\-bi)  =  A  C^  [645]. 

2.  a  —  'bi=-\-a-]-{—'bi). 

.'.  Vector  {a-hi)  =  A  C^  [645]. 

3.  —a-\-hi=—a-\-{-\-hi). 

.'.  Vector  {-a^hi)  =  AC^  [645]. 

4.  — a  —  hi=— a-{-{—  hi). 

.-.  Vector  {--a-hi)  =  AC^  [645]. 

647.  The  vectors  of  the  two  terms  of  a  complex  num- 
ber are  called  the  components  of  the  vector  of  the  number. 

Thus,  A  B  and  B  C^  are  the  components  of  A  C^. 

648.  The  length  of  the  vector  of  a  complex  number  is 
called  the  modulus  of  the  vector,  or  the  modulus  of  the 
complex  number,  and  is  equal  to  the  square  root  of  the 
sum  of  the  squares  of  the  lengths  of  the  components. 

Thus,  mod.  {a-\-hi)  =  mod.  (a  — hi)  =  mod.  {—a-\-hi) 
=  mod.  {—a  —  hi)  =  Va^  +  h\ 

649.  The  direction  of  the  vector  of  a  complex  number 
is  determined  hy  the  angle  which  the  vector  makes  with 


358  ADVANCED  ALGEBRA. 

its  horizontal  component,  which  angle  is  called  the  ampli- 
tude of  the  yector. 

Thus,  the  amplitude  of  the  vector  A  Ci  is  the  angle 
Ci  A  B,  which  for  distinction  will  be  represented  by  u)^ ; 
the  amplitude  of  A  Cz  is  Cz  A  B',  represented  by  Wg ;  the 
amplitude  of  A  G^  is  G^  A  B\  represented  by  6)3 ;  and  the 
amplitude  of  A  C4  is  G^AB,  represented  by  0)4. 

650.  It  is  equally  accurate,  and  sometimes  more  con- 
venient, to  define  the  amplitude  of  a  vector  as  the  angle 
included  between  the  vector  and  the  vector  +  a,  measured 
in  a  positive  direction  from  the  vector  +«. 

Thus,  the  amplitude  of  A  Gz  is  Gz  A  B,  The  ampli- 
tude of  A  C3  is  the  reflex  angle  BAGs,  described  by  re- 
volving A  B  about  A  in  Si  positive  direction  until  it  coin- 
cides with  A  C3.  The  amplitude  of  A  C4  is  the  reflex 
angle  BAG4,. 

651.  A  vector  and  its  components  may  be  constructed 
from  its  modulus  and  amplitude  as  follows  : 

1.  Draw  an  indefinite  horizontal  line,  and  select  some 
point  in  this  line  for  the  origin  of  the  vector. 

2.  At  the  origin,  deflect  an  angle  with  a  protractor 
equal  to  the  amplitude  of  the  vector,  and  in  the  proper 
position. 

3.  Lay  off  from  the  origin,  on  the  deflected  side  of  the 
angle,  from  a  scale  of  equal  parts,  the  modulus  of  the 
vector.     The  vector  is  then  determined. 

^.  At  the  extremity  of  the  vector  let  fall  a  perpendicu- 
lar to  the  horizontal  line.  This  perpendicular  and  the 
part  of  the  horizontal  line  intercepted  between  the  origin 
and  the  foot  of  the  perpendicular  will  he  the  components 
of  the  vector.  Their  lengths  may  he  ohtained  hy  actual 
measurement  on  the  scale  of  equal  parts. 


COMPLEX  NUMBERS. 


359 


Problems. 
652.    To  find  the  sum  of  two  complex  numbers,  graphically. 


the  sum  of  a-\-hi  and 


+  a 


Let  it  be  required  to  find 
—  c-\-d  i. 

1.  Add  —  c-\-di  to  a -{-hi, 
graphically. 

Solution  :  Construct  a  +1)i.  Its 
vector  is  A  C.  At  its  extremity,  (7, 
construct  —c-^di.  Its  vector  is  C  E. 
Join  A  with  E.  AE  is  the  vector  of 
the  sum  [645].  V(6  +  df  +  (a-  cf 
is  its  modulus,  and  the  angle  EA  B  its 
amplitude. 

Proof :  Add  —c  +  di  to  a  +  bi 
algebraically,  and  construct  the  sum. 
Thus, 

{a  +  bi)  +  {—c  +  di)  = 

{a  —  c)  +  {b  +  d)  i. 

Construct  AB  =  +a,  B  C  =  —c ; 
then  AC  =  a  —  c.  At  C erect  CE  = 
{b  +  d)  i.  Join  A  with  E.  The  vector 
A  E  will  be  identical  with  AE  in.  the 
preceding  diagram ;  therefore,  the  solu- 
tion is  correct. 

2.  Add  a-\-hi  to  —c-\-di. 
Solution :    Construct    —c  +  di. 

Its  vector  is  A  C.  At  its  extremity, 
C,  construct  +  a  +  bi.  Its  vector 
is  CE.  Join  A  with  E.  The  vector 
A  E  will  be  identical  with  the  vector 
AE  in  the  first  case,  which  proves 
the  commutative  law  of  addition 
graphically  as  applied  to  complex 
numbers. 

Exercise. — Find  graphically  the  sum  of  : 

1.  a -{-hi  and  c-\-  di  4.  —  a  —  hi  and  —  c  —  di 

2.  a  — hi  and  c  —  di  5.  a -{-hi  and  a  — hi 

3.  a -{-hi  and  c  —  di  6.  0-{-hi  and  0  —  di 


-\-di 


+  bi 


360 


ADVANCUn  ALGEBRA. 


653.    To  multiply  a  complex  number  by  a  rational  num- 
ber, graphically. 

Let  it  be  required  to  multiply  a-^-ii  hy  —  c. 

Solution : 

Construct  a-\-hi.  Pro- 
long its  vector  JL  (7  to  Z), 
making  AD  z=  G  y.  AG. 
Revolve  A  D  about  A 
through  180°,  then  AD 
is  the  vector  of  —  c  times 
a  +  &  *. 

Exercise. — Construct  the  vectors  of  : 

1.  («  +  ^  t)  X  c  4.  {—a  —  b  i)  X  c 

2.  {a  —  bi)Xc  b.  {a  —  h  i)  X  (—  c) 

3.  {—a-\-bi)Xc  6.  {— a -^  ii)  X  (— c) 


664.    To  multiply  a  complex  number  by  a  simple  im- 
aginary number,  graphically. 


Let  it  be  required  to  multiply 
Solution :    Construct  —  a  +  bi. 

making  AD  =  c  x  AG.    AD 

is  the  vector  of  c  times  —  a 

+  b  i.      Revolve    A  D    about 

A  through  an  angle  of    90° 

clock-wise  [641],  or,  which  is 

the  same,  draw  AD  —  AD 

and    perpendicular    to    A  D. 

A D   is  the  vector  of    —  ci 

times   —  a  +  bi.     DAA!    is 

its  amplitude. 

Exercise. — Construct  the  vectors  of 


a -{-hi  by  —  ci. 
Prolong  its  vector  AG  to  D^ 

d' 


1.  {a-{-li)  Xci 

2.  (—a  —  bi)  X  ci 

3.  {a  —  bi)  X  (—ci) 

4.  (a  +  bi)  X  (—ci) 

5.  (—a  +  bi)  Xci 


6.  (a  —  b i)  Xci 

7.  (—a  —  bi)x  (—ci) 

8.  (0  —  bi)  X  (—ci) 

9.  (0  +  b i)  Xci 

10.  (O  +  bi)  X  (—ci) 


COMPLEX  NUMBERS. 


361 


^x 


655.  To  multiply  a  complex  number  by  a  complex  num- 

ber, grraphically. 

Let  it  be  required  to  multiply  —  a-{-di  by  c  —  di. 

Solution:  The  vector  of 
the  sum  of  c  times  —a  +  bi 
and  —  di  times  —a+bi 
is  required.  Construct  —  a 
+  b  i.  Its  vector  is  A  C. 
Prolong  AC  to  D,  making 
AD  =  c  times  AC;  then 
AD  is  the  vector  of  c  times 
—  a  +  bi.  Prolong  ^ Z>  to 
E,  making  DE  =  d  times 
A  C,  and  revolve  D  E  about 
D  through  an  angle  of  90° 
clock-wise,  then  is  DE'  the 
vector  of  —di  times   —a 

+  bi  constructed  at  the  extremity  of  A D.    Join  A  and  E'.    A E'  is 
the  vector  of  the  sum  of  c  times  —a  +  bi  and  —di  times  —a  +  bi. 

Exercise.— Multiply  graphically  : 

1.  a-\-bihyc-\-di  4.  —  a-^-bi  hj  c-\-di 

2.  a  —  hi  hj  c-\-di  b,  — a  —  lihj— c-\-di 

3.  a  —  bihjc  —  di  6.  —  a  — bi  hy  —  c  —  di 

General   Principles. 

656.  1.  The  sum,  the  difference,  the  product,  and  the 
quotient  of  two  complex  numbers  are,  in  general,  complex 
numbers. 

For,  1.  {a^bi)-\r(c^-di)  =  {a^c)-\-{b-\-d)i. 

2.  {a-^bi)-(c  +  di)  =  {a-c)-{-{b-d)i. 

3.  {a-{-b i)  {c  +  di)  =  ac-}-bci-{-adi  —  bd 

=  {ac  —  bd)-\-{bc-i-ad)i, 
a-{-bi  _  {a-\-  bi)  (c  —  di) 
c-\-di  ~  (c-\-  d  i)  {c  —  d  i) 


4. 


{a  c  -\-b  d)  -\-  {b  c  —  a  d)  i  _  ac-\-bd 


+ 


(be  —  ad\. 


862  ADVANCED  ALOEBRA. 

667.  2.  The  sum  and  the  product  of  two  conjugate 
complex  numbers  are  real. 

Eor,  1.  {a-\-hi)-^{a  —  'bi)  z=%a. 

2.  {a-\-di)(a-bi)      =a^  +  bK 

Scholium,  a^  +  b^  is  the  square  of  the  modulus  of 
±a-\rbi  and  of  ±  a  —  bi,  and  is  called  the  norm  of 
each.     Therefore, 

Cor, — The  product  of  two  conjugate  complex  numbers 
equals  their  norm. 

668.  8.  The  norm  of  the  product  of  two  complex  num- 
bers equals  the  product  of  their  norms. 

For,  norm  {a-\-b  i)  {c  +  di) 

=  norm  {{ac —  bd)-\- (ad-\-hc)i] 
-{ac-bdf-\-{ad-\-b cf  [657,  Sch.] 
=  a^  c"  -\-b^  d^  -^  a^  d^  -^-b^  d" 

=  norm  {a-\-b  i)  multiplied  by  norm  {c-\-  d  i). 

Cor, — The  modulus  of  the  product  of  two  complex  num- 
bers equals  the  product  of  their  moduli. 


669.   Jf.  If  a-^bi  =  0,  then  a  =  0  and  b  =  0. 

For,  it  a  -\-  b  i  =  0,  bi=  —a  and  —b^  =  a^; 
whence,  a^-\-P  =  0,  which  is  possible  only  when  a  =  0 
and  b  =  0. 

Cor, — If  a  complex  number  vanishes,  its  modulus  van- 
ishes ;  and  conversely,  if  the  modulus  vanishes,  the  complex 
number  vanishes. 

660.  S.  Ifa-^bi  =  c-\-di,  then  a  =  c  and  b  =  d. 

For,  if  a-\-bi  =  c-\-di,  (a  — c) -{- {b  —  d)i  =  0  ; 
whence,  a  —  c  =  0  and  b  •—  d  =  0  [P.  4], 
and         a  =  c  and  b  =  d. 


COMPLEX  NUMBERS.  363 

661.       Problem.    To  find  the  value  of  e*+^*. 

Solution :  Assuming  the  exponential  law  of  multiplication  [275,  P.], 
and  Formula  (G),  Art.  599,  sufficiently  general  to  include  imaginary- 
exponents;  then 

(  2/H"2       y^i^       y*i*         ^    ) 

e«+y»  =  e'  X  ev'  =  e'  <l+yi+  ^  +  ^  +  ^  +  etc. j- 


v^       v*       y^ 
662.  The  expression  1~J2"I"|4~^  +  ^*^*  ^^  called 

cosine  y,  and  is  written  cos.  y. 

ti^       if*       ip 
The    expression    V  ~  f^  ~\-  h. vi  "^  ^^^'    ^^    called 

sine  y,  and  is  written  sin.  y.     Therefore, 

e^'  =  cos.  y-\-i  sin.  y.  (B) 


663.  Resume  the  equations 

ifi       ^       -^/^ 
cos.y  =  l-|  +  |-|+etc.  (1) 

sin.  3/  =  2/-|-  +  |--|-  +  etc.  (2) 

e''*  =  COS.  y  -\-i  sin.  «/.  (3) 

Put  —y  tor  y  in  (1),  (2),  and  (3),  then 
COS.  (-  2/)  =  1  -  ||  +  1^  -  ^  +  etc.  =  COS.  y  (4) 

sin.  {-  y)  =  -  y  +  y-  -U-  +  y        etc.  =  -  sin.  y  (5) 

e-**  =  COS.  (—  y)  +  i  sin.  (—  y)  —  cos.  y  —  i  sin.  y  (6) 

Multiply  (3)  by  (6), 

1  r=  (cos.  yf  —  i^  (sin.  y)^  —  cos.^  y  +  sin.'  y  (C) 

Note,    cos.*^  y  denotes  (cos.  y)^  and  sin,^  y  denotes  (sin.  y)'. 


Cor,     sin.  y  =  vl  —  cos.^  y  ^ 
COS.  y  =  a/1  —  sm.^  y. 


364  ADVANCED  ALGEBRA. 

664.  Put  ny  ioT  y  in  (B) ;  then 

e"*'*  =  COS.  ny  -\-i  sin.  n  y, 
Eaise  (B)  to  the  n'Cd  power ;  then 

e""*  =.  (cos.  y  -\-i  sin.  yf. 
.-.  COS.  ny  -\-i  sin.  ny  =  (cos.  y  -\-i  sin.  «/)"  (D) 


665.  Let    n  =  %  in  (D) ;  then 

COS.  2  ?/  +  ^  sin.  %y  •=■  (cos.  «/  +  *'  sin.  yY 
=  cos.^  «/  —  sin.^  y  -\-^  (sin.  y  cos.  y)  i. 
.*.  cos.  2«/  =  cos.^  y  —  sin.^  ?/  [660]  (E) 

sin.  2y  =  2  sin.  y  cos.  «/  [660]  (F) 

666.  Put  x-\-y  tor  y  in  (B) ;  then 

g(x  +  y)i    _.    gxi   ^   gyi  _  gQg^  ^^  _|_  1^)  _|_  i;  gin^   (^  _j_  ^)^ 

But    e'*  .  y'^  =  (cos.  a;  +  ^  sin.  ir)  (cos.  y  ^i  sin.  ?/)  (B) 
COS.  X  COS. «/  —  sin.  x  sin. «/  +  *  (sin.  x  cos.  ly  +  cos.  x  sin.  ?/) 
'.  COS.  {x-\-y)  =  cos.  a;  cos.  y  —  sin.  a;  sin.  y  [660]         (G) 
sin.  (^  +  ?/)  =  sin.  x  cos.  ?/  +  cos.  x  sin.  ?/  [660]         (H) 


Graphical  Representation  of  sin.  y  and  cos.  y. 

667.  It  is  evident  that  all  the  conditions  expressed  in 
the  equation  sin.^  y  +  cos.^  «/  =  1  will  be  satisfied  by  as- 
suming 1  as  the  modulus  of  a  vector  whose  amplitude  is 
the  variable  angle  y  and  whose  components  are  sin.  y  and 
COS.  y.  But,  to  make  this  expression  conform  to  the  nu- 
merical values  of  sin.  y  and  cos.  y  as  expressed  in  Art.  662, 
y  must  be  taken  to  represent  the  number  of  vector  units 
in  the  arc  which  measures  the  amplitude,  and  sin.  y  as 
the  vertical  and  cos.  y  as  the  horizontal  component  of 
the  vector ;  for  in  this  way  only  would  sin.  y  =  0  and 
cos.  y  =  l  when  y  =  0, 


COMPLEX  NUMBERS. 


365 


668.  The  ratio  of  sin.  y  to  cos.  y  is  called  tangent  y, 
and  is  written  tan.  y.  It  may  be  expressed  graphically 
as  follows  : 

Let  BC  =  y,  CE 
=  sin.  y,  and  AE  = 
cos.  y.  At  B  draw  an 
indefinite  tangent  to 
the  circle.  Prolong  the 
yector  AC  until  it 
meets  the  indefinite 
tangent  at  i>.  B  D  will 
be  tan.  y.  For,  from 
the  similar  triangles 
DAB  and  CAB  we 
have  BD'.CE  '.'.AB'.AE',  or, 

B  D  :  sin.  y  ::1  :  cos.  y  ;  whence, 
sin.  y 


BD  = 


tan.  y. 
sin.  y,  and  A  L  ^  cos.  «/.     The  tri- 


ces, y 

If  BF=zy,  FL 
angles  A  FL  and  B  A  K  will  be  similar,  and  B  K=^  tan. «/. 

If  B  F  G  =  y,  then  G  L  =.  sin.  y,  and  ^  Z  =  cos.  ?/. 
The  triangles  GAL  and  DAB  wiU  be  similar,  and  D  B  =:^ 
tan.  y. 

If  ^ i^ZT  =  y,  then  HE  =  sin.  y,  and  AE  =  cos.  «/. 
The  triangles  KAB  and  -ST^  ^  will  be  similar,  and  KB 
—  tan.  y. 

Scholium, — So  long  as  y  <  \  it  [90°],  sin.  y  and  cos.  y 
are  positive  ;  hence,  tan.  y  (B  D)  is  positive.  When  y  > 
i  TT  but  <  IT,  sin.  y  is  positive  and  cos.  y  negative  ;  he^ice, 
tan.  y  (B  K)  is  negative.  When  y  >  tt  tut  <  f  t,  sin.  y 
is  negative  and  cos.  y  negative  ;  hence,  tan  y  {D  B)  is  posi- 
tive. When  y  >  i  TT  but  <  2  v,  sin.  y  is  negative  and 
cos.  y  is  positive;  hence,  tan.  y  {KB)  is  negative. 


CHAPTER  XI. 
THEORY    OF    FUJVCTIOJVS. 


Definitions. 


669.  A  quantity  whose  value  changes,  or  is  supposed  to 
change,  according  to  a  definable  law,  is  a  definite  variable, 
or  simply  a  variable. 

670.  A  variable  whose  law  of  change  is  not  dependent 
upon  that  of  another  variable  is  an  independent  variable, 

671.  A  variable  whose  law  of  change  is  dependent  upon 
that  of  another  variable  is  a  dependent  variable,  and  is 
called  a  function  of  that  variable. 

Hence  it  is,  that  any  expression  containing  a  variable 

is  a  function  of  that  variable  [562]. 

• 

672.  Any  law  of  change  may  be  imposed  upon  an  inde- 
pendent variable ;  but,  when  it  is  once  imposed,  the  law 
of  change  of  any  function  of  the  variable  becomes  de- 
termined. 

673.  The  simplest  treatment  of  functions  of  a  single 
variable  is  that  in  which  the  variable  is  supposed  to  in- 
crease or  decrease  uniformly  by  equal  increments,  finite  or 
infinitely  small. 

674.  A  function  is  said  to  be  continuous  so  long  as  an 
infinitely  small  change  in  the  independent  variable  pro- 
duces an  infinitely  small  change  in  the  function,  and  dis- 


THEORY  OF  FUNCTIONS.  367 

continuous  when  an  infinitely  small  change  in  the  inde- 
pendent variable  produces  a  finite  or  infinitely  great  change 
in  the  function. 

Ulustration. — Thus,   the  function    - — —    assumes  all 

values  between  +  1  and  +  oo  as  a;  assumes  all  values  be- 
tween 0  and  + 1,  and  is,  therefore,  continuous  from  0  to 
+  00 ;  but,  as  the  value  of  x  continues  to  increase  from  a 
quantity  infinitesimally  less  than  +  1  to  a  quantity  infini- 
tesimally  greater  than  +  1,  or  takes  an  infinitely  small  step 
across  + 1,  the  function  takes  a  leap  through  the  whole 
gamut  of  numbers  from  +  "^  to  —  oo ,  and  is,  therefore, 
discontinuous  between  these  values. 

675.  So  long  as  a  function  increases  in  value  as  the 
independent  variable  increases  in  value,  and  hence,  too, 
decreases  in  value  as  the  independent  variable  decreases  in 
value,  it  is  an  iticreasing  function  ;  but  when  it  decreases 
in  value  as  the  independent  variable  increases  in  value, 
and,  hence,  increases  in  value  as  the  independent  variable 
decreases  in  value,  it  is  a  decreasing  function. 

Ulustration. — Let  y  =f(x)  =  x^  —  4,x-\-3. 

Assign  values  to  x  and  calculate  the  corresponding 
values  of  y  by  synthetic  division  [106],  you  will  obtain 
results  as  follows  : 

For  a;  =  -  3,  -  2,  -1,      0,  +1,  +2,  +3,  +4,  +5 
y  =  +24,  +15,  +8,  +3,      0,  -1,      0,  +3,  +8 

Here  y  decreases  from  +  24  to  —  1  as  a;  increases  from 
—  3  to  +2,  and  is,  therefore,  a  decreasing  function  be- 
tween these  values  of  x  ;  and  it  increases  from  —  1  to  +8 
as  X  increases  from  +2  to  +5,  and  is,  therefore,  an  in- 
creasing function  between  these  values  of  x. 

676.  The  maximum  value  of  a  function  is  the  value  at 
which  the  function  changes  from  an  increasing  to  a  de- 
creasing function. 


368  ADVANCUn  ALGEBRA. 

677.  The  minimum  value  of  a  function  is  the  value  at 
which  the  function  changes  from  a  decreasing  to  an  in- 
creasing function. 

678.  The  maxima  and  minima  values  of  a  function  are 
often  called  the  turning  values  of  the  function. 

679.  A  turning  value  of  a  function  may  be  a  finite 
constant,  zero,  or  infinity. 

niustrations.— 1.  Take  ^^  =  /(i»)  =  3  +  (4  -  xf. 

As  X  increases  from  0  to  4,  «/  decreases  from  19  to  3 ; 
and  as  x  continues  to  increase  from  4  to  oo ,  y  increases 
from  3  to  00.  Therefore,  3  is  a  turning  value  (a  mini- 
mum) of  y. 

2.  Take  y  =  {a-  xf. 

As  X  increases  from  0  to  a,  y  decreases  from  a^  to  0 ; 
and  as  x  continues  to  increase  from  a  to  co ,  y  increases 
from  0  to  00 .     Therefore,  0  is  a  turning  value  of  y. 

As  X  increases  in  value  from  0  to  + 1,  (1  —  xy  decreases 
from  1  to  0,  and  y  increases  from  1  to  oo ;  and  as  x  con- 
tinues to  increase  from  1  to  oo ,  (1  —  xY  increases  from 
0  to  00 ,  and  y  decreases  from  oo  to  0.  Therefore,  oo  is  a 
turning  value  (a  maximum)  of  y. 

680.  The  limit  of  a  function  is  the  value  of  the  func- 
tion at  which  it  ceases  to  be  continuous. 

Note. — Notice  the  distinction  between  the  meaning  of  the  word 
limit  as  here  used  and  as  used  in  Art.  398.  In  the  latter  sense,  <x> 
would  be  the  limit  of  y  in  illustration  3,  Art.  679,  instead"  of  a 
maximum. 

681.  The  limit  of  a  function  may  be  a  finite  constant, 
zero,  or  infinity. 

Illustrations. — 1.  Take  y  =f(x)  =  2  —  —, 

2 
As  X  increases  from  0  to  oo ,  ~  decreases  from  2  to  0, 


THEORY  OF  FUNCTIONS.  369 

and  y  increases  from  0  to  2  ;  and,  as  x  can  n(#  be  sup- 
posed greater  than  oo ,  y  can  not  become  greater  than  2, 
neither  can  y  begin  to  decrease  at  2.  Therefore,  2  is  the 
limit  of  y. 

2.  Take  y  z=z  f{x)  =  x^{l^  x^). 

As  X  decreases  from  1  to  0,  y  decreases  from  2  to  0 ; 
and  as  x  can  not  be  taken  less  than  0  (negative)  without 
making  y  imaginary,  y  can  not  become  less  than  0,  neither 
can  y  change  from  an  increasing  to  a  decreasing  function 
at  0.     Therefore,  0  is  the  limit  of  y, 

3.  We  have  already  seen  [674]  that  y  —  f(x)  =     __ 

increases  from  1  to  oo  as  a;  increases  from  0  to  +1,  and 
thereafter  becomes  discontinuous.  Therefore,  (»  is  the 
limit  of  y, 

682.  A  function  may  have  two  sets  of  values  approach- 
ing the  same  or  different  limits  for  the  same  set  of  values 
of  the  independent  variable. 

niustrations.— 1.  Take  y^  =/(a:)  =  16  —  a;^ ; 
then  y  =  ±  a/16  -  a;^. 

Here  are  two  values  of  y  for  each  value  of  x,  numeri- 
cally equal  but  opposed  in  sign.  As  x  increases  from  0 
to  4  one  value  of  y  decreases  from  4  to  0,  and  the  other 
increases  from  —  4  to  0.  It  x  becomes  infinitesimally 
greater  than  4  both  values  of  y  become  imaginary.  There- 
fore, 0  is  the  limit  of  both  values  of  y. 

2.  Take?/2  =/(^)  =  4a:; 
then  «^  =  ±  2  Vx. 

Here,  again,  are  two  values  of  y  for  each  value  of  x. 
As  X  increases  from  0  to  +  oo ,  one  value  of  y  increases 
from  0  to  -f  °°  and  the  other  decreases  from  0  to  —  oo ; 
and  as  x  can  not  be  supposed  greater  than  -\-  co ,  +  °o  ^s 
the  limit  of  one  value  of  y  and  —  oo  the  limit  of  the  other 
value. 

683.  The  limit  of  an  increasing  function  is  a  superior 


370  ADVANCED  ALGEBRA. 

or  maximum  limit ;  that  of  a  decreasing  function  an  iw- 
ferior  or  minimum  limit. 


Graphical  Representation  of  Functions  of  a 
Single  Variable. 

684.  Every  function  of  a  single  variable  may  be  ap- 
proximately represented  by  a  line,  straight  or  curved, 
called  the  graph  of  the  function. 

Method. — Let  y  =  f{x).  Assign  successive  values  to  x 
and  calculate  the  corresponding  values  of  y.  Construct 
two  indefinite  straight  lines  intersecting  each  other  at  right 
angles,  one  running  right  and  left  and  the  other  up  and 
down  from  their  intersection.  These  are  the  axes  of  refer- 
ence. The  first  is  the  a;-axis  and  the  second  the  i^-axis, 
and  their  intersection  the  origin.  Regard  distance  right- 
ward  from  the  y-axis  positive^  and  distance  leftward  nega- 
tive ;  distance  upward  from  the  a^-axis  positive,  and  dis- 
tance downward  negative. 

Assume  a  fixed  length  as  a  unit  of  scale,  and  lay  off  on 
the  a;-axis  from  the  origin  the  successive  values  of  x  based 
on  this  scale,  and  at  the  extremity  of  each  x  value,  and  on 
a  line  parallel  to  the  y-axis,  lay  off  the  corresponding  values 
of  y.  Thus  will  be  located  a  series  of  successive  points ; 
draw  a  continuous  line  through  these  points ;  it  will  be 
the  graph  of  the  function,  and  its  accuracy  will  depend 
upon  the  nearness  to  each  other  of  the  successive  values 
of  X  taken,  the  relation  of  the  unit  of  scale  to  that  of  x 
and  y,  and  the  correctness  of  the  instruments  used  in 
plotting. 

niustrations.— 1.  Take  y  =f{x)  =  x^  —  20  x^ -\- QL 
Assign  special  values  to  x  and  calculate  the  correspond- 
ing values  of   y  by  synthetic  division   [106].     You  will 
readily  derive  the  following  table  of  values  and  make  the 
following  plot : 


FUNCTIONS  OF  A  SINGLE   VARIABLE.         371 


X 

y 

0 

64 

1 

44 

2 

0 

3 

-35 

4 

0 

5 

189 

00. 

00 

-  1 

44 

-  2 

0 

-  3 

-35 

-  4 

0 

-  5 

189 

—  00 

00 

PLOT. 

+  2/ 


Observa- 
tions.— 1.  The 
curved  line 
on  the  plot  is 
the  graph  of 
the  function. 

2.  The  unit 
of  scale  used  in 
plotting  the 
graph  repre- 
sents 20  units 
of  y  to  one 
unit  of  X. 


+ 

200 

+ 

180 

-f- 

160 

1 

+ 

140 

+ 

120 

+ 

100 

+ 

80 

+^ 

v60 

/ 

/. 

.oN 

\ 

/ 

+ 

20 

\ 

/ 

4 

-3 

-2 

A 

0 

A 

+2 

+3 

u 

\ 

/ 



20 

\ 

y 

V 

/ 



40 

\ 

/ 

__ 

60 

-y 


3.  The  graph  exhibits  three  turning  values  of  the  function ;  two 
minima  at  the  points  (ic  =  3,  y  =  —  35)  and  (—  3,  —  35),  and  one 
maximum  at  the  point  (0,  64). 

4.  When  a:=+oo,  2/r=+oo,  and  when  a;=— oo,  y  =  +  co. 
The  graph,  like  the  function,  descends  from  (—  oo ,  +oo)  to  (—3,  —35), 
then  ascends  from  (—  3,  —  35)  to  (0,  64),  then  descends  from  (0,  64)  to 
(3,  —  35),  then  again  ascends  from  (3,  —  35)  to  (+  oo ,  +  oo ).  It  is  a 
continuous  graph  from  beginning  to  end. 

5.  At  a?  =  2,  4,  —  2,  and  —  4,  the  graph  crosses  the  ir-axis,  exhibit- 
ing the  fact  that  for  these  values  of  x,  y  =  f(x)  =  x^  —  20  x^  +  64t  —  0. 
The  values  of  x  that  render  f(x)  =  0  are,  however,  the  roots  of  the 
equation  f(x)  =  0;  therefore,  the  values  of  the  roots  of  f(x)  =  Q  may- 
be approximately  found  even  if  incommensurable,  by  plotting /(a;)  =  y 
and  determining  with  a  scale  of  equal  parts  where  the  graph  crosses 
the  a^^axis. 


372 


ADVANCED  ALGEBRA. 


2.  Plot  y  =  ±  Va^-x  =  ±  Vx(x+l){x -1). 
The  following  table  of  values  may  readily  be  obtained 


-^07 


+ 

y 

3 

PLOT. 

/ 

+ 

3.5 

/ 

+ 

2 

/ 

+ 

1.5 

/ 

+ 

1 

/ 

/ 

+ 

.5 

/ 

-1 

/d 

■^ 

0 

+.5 

/ 

-fl.5 

+2 

+2.5 

V 

^ 

.5 

\ 

1 

\ 

1.5 

\ 

y 

2 

\ 

2.5 

\ 

_ 

3 

\ 

\ 

+3 


Observations. — 1.  The  graph  consists  of  two 
branches  between  the  points  (—1,  0)  and  (0,  0), 
symmetrical  with  respect  to  the  a;-axis.  These 
branches  are  confluent  at  the  points  mentioned. 

2.  The  graph  is  discontinuous  for  all  values 
of  X  antecedent  to  —  1,  counting  from  a;  =  —  oo , 
and  also  for  all  values  of  x  between  0  and  +  1. 

3.  The  graph  again  consists  of  two  branches, 
symmetrical  with  respect  to  the  a;-axis,  for  all 
positive  values  of  x  greater  than  +  1. 

4.  The  limits  of  the  first  branch  are  at  (—1,  0) 
and  (0,  0) ;  the  limits  of  the  second  branch  are  also 
at  (—1,  0)  and  (0,  0) ;  the  limits  of  the  third  branch 
are  at  (+  1,  0)  and  (+  oo ,  +  oo ) ;  and  the  limits  of 
the  fourth  branch  are  at  (+  1,  0)  and  (+  oo ,  —  oo ). 


+  x 


X 

y 

0 

0 

+  <  +  l 
1 
1-5 

±1-4 

2 

±2-4 

2-5 

±3-6 

3 

±4-9 

-1 

0 

-<-l 

V- 

-•2 

±•44 

-•4 

±•58 

-•5 

±•61 

-•6 

±  ^62 

-•8 

±•53 

FUNCTIONS  OF  A  SINGLE   VARIABLE. 


373 


5.  The  first  branch  has  a  turning-point  (maximum)  somewhere 
between  (—  '5,  +  -61)  and  (—  '8,  +  'SS).  The  second  branch  also  has  a 
turning-point  (minimum)  between  (—  '5,  —  '61)  and  (—  '8,  —  '53). 

6.  The  branches  of  the  graph  meet  the  a;-axis  when  a:  =  0,  +1, 
and  —  1.    These  values  are,  therefore,  the  roots  of  f{x)  =  x^  —  x=.0, 

3.  Plot  2/2  =  a;3_9^^24a;^16^ 

or      y   =  ±  v^ 

y 


X 

y 

<1 

V- 

1 

0 

1-5 

±  1-76 

2 

±3 

3 

±1-41 

4 

0 

5 

±2 

6 

±4-47 

7 

±7-34 

+  00 

±00 

PLOT. 


16. 


Questions. — 

1.  How  many 
branches  has 
this  graph  ? 

2.  How  many  ^/ 
turning-points? 

Locate     and   _ 
name  them. 

3.  What   are 
the  roots  of  the    "~ 
function    x^  — 
9x2 -h  24a; -16    _ 
=  0! 

4.  Between 
what  limits  are    "" 
the  branches  of 
the  graph  con-    _ 
tinuous  ? 

5.  Where    is 
the  function  an    ~~ 
increasingfunc- 
tion  and  where 

a  decreasing? 


6 

/ 

5 

/ 

4 

/ 

/ 

3 

/ 

2 

/ 

1 

f 

\, 

/ 

0 

i 

1 

2 

•\ 

/ 

5 

6 

1 

\ 

/ 

\ 

2 

V 

/ 

\ 

3 

\ 

4 

\ 

5 

\ 

\ 

6 

\ 

\ 

y 

\ 

374  ADVANCED  ALGEBRA. 

EXERCISE    96. 

Plot  and  discuss  the  following  functions  : 
(Use  paper  ruled  in  squares,  called  l)lotting-paper.) 

1,  y  =^x-\-Q  6,  1/^  =  a;^ 

2.  y  =Sx  1.  y^  =:x^{x—  1) 

3,  y  =81a;-3  8.  y  =a^ -'8x^-\-20x -10 

4.  y^  =  4:X  9.  y  =3x-\-lSx^  —  2a^ 

b.  y^  =  16-x^  10.  y^  =  3^-{-ds^-5x-20 


Differentials  and  Derivatives  of   Functions. 

Definitions. 

685.  The  limit  of  the  ratio  of  the  increment  of  a  func- 
tion to  the  increment  of  the  independent  variable  pro- 
ducing the  increment  of  the  function,  when  the  limit  of 
the  increment  of  the  independent  variable  is  zero,  is  called 
the  derivative  of  the  function. 

Thus,  if  we  let  y  =/(a;),  and  represent  the  increment 
of  a;  by  A  a;  and  the  corresponding  increment  of  y  hj  Ay, 

then  will  lim.  ( — -  ]  =  the  derivative  of  the  function. 


m 


686.  The  limit  of  the  increment  of  the  independent 
variable  is  called  the  differential  of  the  independent  vari- 
alle,  and  is  represented  hy  dx\  and  the  limit  of  the  incre- 
ment of  the  function  is  called  the  differential  of  the  func- 
tion, and  is  represented  by  dy. 

dy 


:  -■  (fl) 


Therefore,  x^^x.  ,         ,  _   , 

Q     ax 


Notice,    dy  and  dx  represent  single  quantities  (differentials)  and 
are  not  equivalent  to  d  x  y  and  d  y.  x. 


DIFFERENTIALS  AND  DERIVATIVES.  375 

Ulustration. — 

Let  y  =  7^  (1) 

then,  y-\-  /\y  =  {x -^ /^xf  =  x^ -{-%x(Ax)  -\-{^xf  (2) 

Subtract  (1)  from  (2),  ^y  = '^xi^x)  +  {^xf  (3) 

Divide  by  a  x,  -^  =  2a;  +  A  a;  (4) 

A  X 

.-.        Lim.fA|)  ^lim.(2a:  +  Aa;)  [401,P.]      (5) 

.:    -r^  =  2x       =  the  derivative  of  a;', 
dx 

and  dy  =  2xdx  =  the  differential  of  a;'. 

687.  The  differential  of  a  function  equals  the  derivative 
of  the  function  multiplied  ly  the  differential  of  the  inde- 
pendent variable, 

688.  The  derivative  of  a  function  equals  the  differential 
of  the  function  divided  by  the  differential  of  the  independ- 
ent variable. 

689.  //  the  differential  of  a  function,  and  hence,  too, 
the  derivative  of  a  function,  is  positive,  the  function  is  an 
increasing  one;  if  negative,  a  decreasing  one. 

Principles. 

690.  Let  y  =f{x)  =  x%  (1) 
then,  y-]-  Ay  =  {x-\-  Axy 

=  x''-\-nx''-'^  .Ax-{-A.{AxY    (2) 
in  which       A  = 
n{n-l)  ^_, _^  n(n-l)in-2)  ^_3 .  ^  ^  +  ^tc.  [593]. 

Subtracting  (1)  from  (2),  Ay  =  nx^-^  .  Ax  +  B  .{Axf        (3) 

A  tJ 

Dividing  by  A  x,  — ~  =  wa;*-i  +  -B  .  A  a;  (4) 

A  X 

Lim.  (  ^^  )  =  lim.  (na;«-i +  5.  A  a;)^  ^_o 

.'.    ~  =  nx^-\  since  Km.  ^  =  a  finite  constant  [582],  and 

lim.  A  a;  =  0.  (5) 

.'.    dy  =  nx^-'^dx.    Therefore, 


376  ADVANCED  ALGEBRA. 

JPrin,  1. — The  differential  of  a  variable  with  a  con- 
stant exponent  equals  the  continued  product  of  the  expo- 
nent, the  variable  with  its  exponent  diminished  by  unity, 
and  the  differential  of  the  independent  variable. 

Ulustrations. — 

1.  d{a^)  =  4.a^dx  2.  d{x)-^=  -^x'^dx 

3.  dia  +  bxy^pia  +  bxy-'^dia  +  bx) 


691.  Let  y  =  ax  (1) 

then,  y -\-  Ay  =  a(x-\-  Ax)  =zax-\-a{Ax)       (2) 
Subtracting,    Ay  =  a{Ax)  (3) 

Dividing,        — -  =  a 

A  X 

Lim.  — -  =  a 

A  X 

whence,  -j^  =  a,  and  dy  =  adx.    Therefore, 

Prin,  2, — The  differential  of  a  constant  times  a  vari- 
able equals  the  constant  times  the  differential  of  the  vari- 
able. 

Thus,  d{dx')  =  d  .  d(x')  =  3  X  6x^ dx  =  15a^dx. 


692.  Let  y  =  ax-\-b  (1) 

then,  y-\-  Ay  =  a{x-{-  Ax)-]-b  = 

ax-\-a(Ax)-\-b  (2) 

Subtracting,    Ay  =  a(Ax) 

Dividing,        -^  =  a;  whence,  ^  =  «» 

and  dy  =  adx.    Therefore, 

I*rin,  3, — The  diffetential  of  a  constant  term  is  zero. 


693.  Let        V  =  f{x),  w  =/'  (x),  and  z  =/"  {x) ;  and 
let  y  =z  v-\-w  —  z  (1) 

then,  y  -\-  Ay  =  v-Yav-^w-\-Aw  — {z-]rAz) 

■=V-\-W  —  Z-\-AV-[-AW—  AZ  (2) 


DIFFERENTIALS  AND  DERIVATIVES.  377 

Subtracting,  Ay  =  Av  +  aw—  a  z  (3) 

T^.    .,.        ,  Ay         A  V         AW         AZ 

DmdmgbyA^,  — =  —  +  —  -—  (4) 

,                    ,.        Ay        T        AV       ,.        AW       ,.        AZ 
whence,        lim.  - — ~  =  lim. h  lim. lim. (5) 

[413,P.].  ^"^  ^^  ^"^  AX  '> 

dy  _dv      dw      d  z 

dx  ~  dx      dx       dx  ^  ' 

whence,  dy  =  dv  +  dw  —  dz.    Therefore, 

Prin,  4, — The  differential  of  a  polynomial  whose  terms 
are  functions  of  the  same  independent  variable  equals  the 
algebraic  sum  of  the  differentials  of  its  terms, 

niustration.     ^  (a;^  +  3  a;^  —  2  a;  +  5) 

=  d{7^)  +  d{^x^)  ^  d(-%x)  -\-  d{h) 

=  da^dx-{-6xdx^2dx=  {Sa^ -\-6x —  2)dx. 


694.  Let  V  =f(x)  and  z  =  f  {x), 

and  y  =  vz  (1) 

then,  y-\-  Ay  =  {v -\-  /\v)  {z -\-  A  z) 

=iVZ-\-V./\Z-{-Z.l\V-\-llV.AZ  (2) 

Subtracting,        'Ay  =  v.Az  +  z.Av-\-Av.Az  (3) 

TV.  .,.      ,  Ay  A  z  A  V        A  V 

Dividing  by  a  x,  — -  =  v  . +  z  . +  — - .  a  z 

^    ^        'ax  ax  ax       ax 

Lim.  — ~  —  lim.  |  v  . )  +  lim.  (  z  . ) 

AX  \        AxJ  \        AxJ 

+  lim.  (-^  .  A  zj 

whence,  dy  =  vdz  +  zdv.    Therefore, 

rrin,  5. — The  differential  of  the  product  of  two  con- 
tinuous functions  of  the  same  independent  variable  equals 
the  swn  of  the  products  obtained  by  multiplying  each  func- 
tion by  the  differential  of  the  other. 

niustration. — 

d{--^7^Xhx^)  =  -dar'xd(6x^)-\-6x^Xd{-Sa^) 

=  {-da^XiX6x-^-{-6xixSxi-3a^)}dx 

=  —  65x^  dx. 


378  ADVANCED  ALGEBRA. 

Cor,    d{vwz)  =  v.d{wz)-\-wz.dv  [694,  P.] 

='Vwdz-\-vzdw-\-wzdv.  [694,  P.]  ;  etc. 

696.  Let       V  =:  f{x),  and  z  =f'(x);  and 

^  -1 

y  =  —  =  vz   ^ : 

z 

then  dy  =  v  ,  d {z-^) -\- z-""  d v  [694,  P.] 
=  —vz~^  dz-\-z~'^dv 
_dv       vdz_zdv  —  vdz 

Therefore, 

Prin»  6, — The  differential  of  a  fraction  whose  terms 
are  continuous  functions  of  the  same  independent  variable 
equals  the  denominator  into  the  differential  of  the  numera^ 
tor  minus  the  numerator  into  the  differential  of  the  de- 
nominator, all  divided  by  the  square  of  the  denominator, 

\yy  y' 

_2xy^dx-'3x^y^dy 
"  ?  • 

696.  Let       y  =  log,  x 

then  y-\-  Ay  =  ]og,{x+Ax)  =  log,  x(l  +  ^j 

=  log,  X  +  loge  (l  +  -^)  [467,  P.  2] 
=  log.a;+  —  -  ^  .  L_^  +  -  .  L_l  -  etc.  [601,  C] 

—  =  -  +  5.  A2:;  in  which  5=_-.-  +  -.--^_  etc.; 

which  for  very  small  values  of  A  a;  is  convergent  [582]. 

.-.    Lhn.  (^\  =  lim.  (^  +B.  ax) 

\AxJi,x  =  0  \X  J^x-0 

whence,  -^  =  —;  and  dy  =  — .    Therefore, 

CuX         X  X 


DIFFERENTIALS  AND  DERIVATIVES,  379 

Prin,  7. — The  differential  of  the  log,  of  a  quantity 
equals  the  differential  of  the  quantity  divided  hy  the  quan- 
tity itself 

Cor, — Since    logio  x  —  m  log,  x  [607,  P.] 

,    .,  .       mdx 

^.(logio^)  =  — ^. 

Illustration. — 

—     ^     J  ^■^^-\-  (x-\-a)dx \  __      1      /2x-\-a\  , 
"^  x-\-a{  x^  \'~  x-\-a\     X     J 


697.  Let  y  =  «*,  in  which  «  is  a  constant, 
then,  log.  y  =^  X  log.  a  [468,  P.] 

d  (log.  y)  =  log.  a  .  dx 

or,  -^  =  log.  a  .  t?a; 

whence,  dy  =  a"  log.  a  .dx.     Therefore, 

JPrin,  8, — The  differential  of  a  constant  with  a  vari- 
able  exponent  equals  the  continued  product  of  the  original 
quantity,  the  logarithm  of  the  constant,  and  the  differential 
of  the  variable  exponent. 

Thus,  d{a-\-l)'^=-d{a-^b)^={a^ bf^ log.  {a  +  b) 

X  d  (x^)  =  i  x-^  (a  +  if^  log.  (a  +  J)  ^  x. 

698.  Problem.    Find  the  differential  of  05*. 

Let  y  —  ^ 

then,      log.  y  =  X  log.  a; 

and  d  (log.  y)  =  x  .  d  (log.  ic)  +  log.  x  ,  dx 

dy            dx   .  ,  J 

or,  — J  =  a;  . 1-  log.  X  .  dx 

X/  X 

whence,     dy  —  xf{\-\- log.  x) dx, 

G 


380  ADVANCED  ALGEBRA. 

EXERCISE    97. 

Differentiate  : 
1.  y=.hao[^ —'^Ix^-^-'ilcx  — d  2.  y  =  5x^z^ -\-z 

^  '  14.  ?/  = 


5'  y—  2^  jg_  y  =  {x-\-  ay  (x  -  h) 


=  /^ 


7.  2^=V5aJ+6  19.  2/  =  3^' 

z.f^%px  20.  y  =  a;^^ 

9.f  =  Za^  21.  2^  =  ic*(a  +  ^^)-i 

•^       '  ,       22.    «/  =  lOge  (a  +  Xf 

^  '  '  23.  y  =  (f  -^  d" 

13.  2/  = 


Vx-\-a  25.  y  =  d'^-' 

Applications. 

EXERCISE    98. 

1.  At  what  rate  is  the  area  of  a  circle  increasing  when 
the  radius  is  6  inches  and  is  increasing  at  the  rate  of  3 
inches  per  second  ? 

Solution :  Let      y  =  the  area,  and  x  =  the  radius ;  then, 
y  z=z  V  x^ 
and  dyz=2'irxdx. 

This  denotes  that  at  any  instant  the  rate  of  increase  of  the  area  is 
2  rr  X  times  as  great  as  the  rate  of  increase  of  the  radius  at  the  same 
instant.  But  when  the  radius  is  6  inches,  it  increases  at  the  rate  of  3 
inches  per  second ;  or,  when  x  =  6  inches,  dx  =  B  inches. 

.'.  dy  =  2ir  X  6  inches  x  3  inches  =  36  ir  square  inches ;  that  is, 
the  area  is  increasing  at  such  a  rate  that,  if  kept  uniform  for  one  sec- 
ond, the  increase  would  amount  to  36  v  square  inches. 


DIFFERENTIALS  AND  DERIVATIVES.  381 

2.  At  what  rate  is  the  area  of  a  square  increasing  when 
the  side  of  the  square  is  4  inches  and  is  increasing  at  the 
rate  of  2  inches  per  second  ? 

3.  The  volume  of  a  sphere  increases  how  many  times  as 
fast  as  its  radius  ?  When  its  radius  is  6  inches  and  in- 
creases at  the  rate  of  1  inch  per  second,  at  what  rate  is  the 
volume  increasing  ? 

4.  At  what  rate  is  the  diagonal  of  a  square  increasing 
when  the  side  of  the  square  is  8  inches  and  is  increasing 
at  the  rate  of  2  inches  per  second  ? 

5.  The  radius  of  a  circle  is  4  inches  and  its  circumfer- 
ence is  increasing  at  the  rate  of  2  tt  inches  per  second.  At 
what  rate  is  the  radius  increasing  at  the  same  instant  ? 

6.  A  boy  approaches  a  tree  90  feet  high  standing  on  a 
level  road  at  the  rate  of  3  miles  an  hour.  At  what  rate 
is  he  approaching  the  top  of  the  tree  when  he  is  220  feet 
from  the  base  ? 

7.  The  diagonal  of  a  cube  is  increasing  at  the  rate  of 
36  inches  per  second,  when  the  side  of  the  cube  is  5  inches 
long.  At  what  rate  is  the  side  increasing  at  the  same 
time  ? 

8.  If  X  increases  at  the  rate  of  '5  per  instant,  at  what 
rate  is  logio  x  increasing  when  a:  =  42  ? 

9.  The  logio  42  =  1-62325.  What,  then,  would  be  the 
logio  42*5,  if  the  increase  were  uniform  ?  How  does  the 
result  compare  with  logio  4:2 '5  as  found  in  the  table  ? 


Successive  Derivatives. 

699.  If  the  derivative  of  f{x)  be  treated  as  a  new  func- 
tion of  X  [/i(^)],  there  may  be  found  from  it  a  second 
derivative  of  f(x)  [/g  {x)"]  in  the  same  way  as  /i  {x)  was 
derived  from  f{x),  and  so  on,  until  a  derivative  is  found 
that  is  independent  of  x  \_f{x^\ 


382  ADVANCED  ALGEBRA. 

Illustration. — 

Let/ (a:)  =  x^ -]-4:a^  - '^a^  ^%7?  -  bx^^ -,  then, 
f^{x)  -  5a:*  +  16a,-3-9a;2  +  4^-5  [693,  P.    688] 
f^{x)  =  20a;3^48^2_i3^_j_4 

f^{x)  =  60a:2_^96^,_i8 
/4(a;)  =  120a;4-96 
f{x,)  =  120 

EXERCISE    99. 

Find  the  first  derivative  of  : 

1.  ic3_4ic2  +  7a;  +  2  4.  (a  +  a:)5  (a  _  a:)3 

2.  (a;  +  2)3  (:z;  -  2)*  5.  {a  +  a:^)  (a  -  t?) 

3.  2:(a;  +  2)  +  a;2(:r  +  3)  6.  («  +  a;)^  ^  («  -  ic)*^ 


Factorization  of  Polynomials  containing  Equal 
Factors. 

700.  Let  f{x)  =  {x-\-  tti)  {x  +  a^)  {x-\-a^)  . . . .  {x-^  a^) 
=  any  polynomial  composed  of  binomial  factors  of  the 
form  of  x-\-a;  then 
/i  (x)  =  {x-\-  ttz)  (x  +  as)  (x-\-a^)  ....(x  +  «„)  + 
{x  +  «i)  (a?  +  %)  {x-{-a^)  ....  (x  +  aj  + 
{x  +  «i)  (:r  +  «2)  (:r  +  «4)  . . . .  (a;  +  o^J  + 
(^  +  «i)  (a;  +  ag)  (^  +  «3) i«^  +  «»  H- 


[694,  Cor.     688]. 

Observations. — 1.  If  no  two  factors  of  f(x)  are  alike,  f{x)  atid 
fx  (a;)  have  no  common  factor. 

2.  If  two,  three,  or  r  factors  of  f{x)  are  equal,  and  all  equal  to 
jc  +  a,  then  will  a;  +  a,  (a;  +  a)*,  or  (a;  +  a)'"^  be  a  common  factor  of 
S{x)  and  /i  (x). 


POLYNOMIALS  CONTAINING  EQUAL  FACTORS.    383 

3.  In  general,  if  f{x)  contains  the  factor  x  -^  a  p  times,  {x  +  h) 

q  times,  {x  +  c)  r  times then  will  {x  +  a)p-'^  {x  +  6)?-*  {x  +  cy-'^ 

....  be  the  H.  C.  D.  of  f(x)  and  /i  (x). 

4.  The  H.  C.  D.  of  f{x)  and  /i  {x)  contains  one  factor  less  of  each 
kind  than  does  f{x). 

701.  Theorem, — Every  polynomial  composed  of  bino- 
mial factors  of  the  first  degree,  some  of  which  are  equal, 
may  he  decomposed  into  factors  containing  no  equal  bino- 
mial factors  of  the  first  degree. 

For,  let  f{x)  be  a  polynomial  composed  of  binomial 
factors  of  the  first  degree,  some  of  which  are  equal,  /i  {x) 
its  first  derivative,  /'  {x)  the  H.  C.  D.  of  f{x)  and  /i  {x), 
and  <\>{x)  the  other  factor  oi  f{x) ;  then, 

1.  <^{x)  will  be  devoid  of  equal  factors  of  the  first 
degree  [700,  4]. 

2.  If  /'  {x)  still  contains  equal  factors  of  the  first  de- 
gree it  may  be  resolved  into  two  factors,  /"  {x)  and  <^'  {x), 
in  which  <ii'  {x)  is  devoid  of  equal  factors  [700,  4]. 

3.  This  process  may  be  continued  until  no  factor  is 
left  that  contains  equal  factors  of  the  first  degree,  which 
will  be  when  the  last  H.  C.  D.  found  is  unity. 

Ulustration.— Eesolve  x''  -\-  x^  —  1%  x^  —  1%  3^  -\-  A:%  x?  -{- 
48  a;^  —  64  ic  —  64  into  factors  devoid  of  equal  binomial  fac- 
tors of  the  first  degree. 

Solution : 

f{x)   =  ic'  +  a;«  -  12a;5  -  12a:*  +  ^oi?  +  48a;2  -  64a;  -  64 

/a  (a;)  =  7a;6  +  62:6  -  60 a:*  -  48a:3  +  144a;2  +  96a;  -  64 

f{x)  =  a:4  _  8a;2  +  16  [158]  =  {x^  -  4)2  =  {x  +2)(a;  +  2)  (a; - 2) (a; - 2) 

4>{x)=f{x)-^P{x)  =  x  +  l 

.-.  /(a;)  =  (a;  +  2)(a;  +  2)(a;-2)(a;-2)(a;  +  l). 

EXERCISE     100. 

Factor : 

1.  x^-^-'^Q^-llx^-VUx-m 

2.  a;«-5ic5  +  a^  +  37tc^-86a;2_j_Y6a;-24 


384 


ADVANCED  ALGEBRA. 


3.  o^-^x'-^Q x^-'^l ic5_|_2i6 2^+243  a^-^^Q  x^-^^  a;-739 

4.  x^""  _  30  a:8  +  345  x"^  -  1900  cc*  +  5040  3?  —  5184 

5.  a;^«-13a;8  4-42ic6-58ar*  +  37a;2_9 


Graphical  Significance  of  /i  (x). 
702.  Let  m/i  be  the  graph  of  y  =f(x). 


Let  P  be  a  point  on  the  graph  whose  co-ordinates  are  x  and  y. 
Let  GE:  =  PR  =  Ax\  then  will  F'R  =  Ay. 

Draw  the  secant  line  P'JPS,  also  the  tangent  line  T'P  T. 
Take  SB  =  1,  and  draw  BC  =  sin.  /S  and  /SC  =  cos.  S. 
Now,  the  triangles  P'  PR  and  jB/SC  are  similar. 

.-.    ^  =  §^=tan.>S'[668]  (1) 

.-.    -^    =:tan. /S  (2) 

AX  ^  ' 

Let  the  point  P'  approach  the  point  P  on  the  graph*  so  as  to 
make  ax  diminish  uniformly;  then  will  the  secant  line  P'  P  S  re- 
volve about  P  and  approach  the  tangent  line  T'  P  T  &s  its  limit,  and 
the  angle  S  will  approach  the  angle  T  as  its  limit. 


Lim.  f -^"i  =  lim.  (tan.  S)i 


or,       :t- 


dy^ 
dx 


tan.  T.    Therefore, 


The  first  derivative  of  a  function  is  equivalent  to  the 
tangent  of  the  angle  which  a  tangent  line  to  the  graph  of 
the  function  makes  with  the  axis  of  abscissas. 


MAXIMA  AND  MINIMA   OF  FUNCTIONS.        385 

Maxima  and  Minima  of  Functions. 

703.  The  maximum  or  minimum  value  of  a  quadratic 
function  may  readily  be  found,  as  follows  : 

Example  1. — What  is  the  maximum  or  minimum  value 
of  x^-\-^x-\-  6,  and  what  value  of  x  will  render  it  a  maxi- 
mum or  minimum  ? 

Solution :  Let       f{x)  =  a;*  +  8a:  +  6  =  w 
Complete  the  square,     ic*  +  8  a:  +  16  =  w  +  10 


Extract  the  \/,  x  +  4  =  ±  ^/nl  +  l6 


Transpose,  x  =  —4±  ^m  +  10 

Now,  w  <  — 10,  else  would  x  be  imaginary. 

m  =  —  10  is  the  minimum  value  of  /  {x). 
But  when  m  —  —  10,  a;  =  —4;   then,  a;  =  —  4  renders  f{x)  = 
a;*  +  8  a;  +  6  =  —  10,  a  minimum. 

Example  2. — What  is  the  maximum  or  minimum  value 
of  8  a;  —  3  2;^  +  9,  and  what  value  of  x  will  render  it  a 
maximum  or  a  minimum  ? 

Solution  :  Let      f{x)  =  8a;-3a;2  +  9  =  m 
Complete  the  square,  9  a;*  —  24  a;  +  16  =  43  —  3  m 
Extract  the  V»  3  a;  -  4  =  ±  \/4S-dm 

Transpose  and  divide,  x  =  -^  ±  -w  \^4d  —  3m 

Now,  3  wi  >  43,  or  m  >  14  -o  ,  else  would  x  be  imaginary. 
1  ^ 

.'.    w  =  14-^  is  the  maximum  value  of  f(x). 

11  1 

But,  when  m  =  14-^ ,  x  =  1-^;  therefore,  x  =  l-^  renders  f(x) 

=  8a;— 3a;*  +  9  =  14-;3- ,  a  maximum. 

o 

Example  3. — Divide  36  into  two  parts  whose  product 
shall  be  the  greatest  possible. 

Solution :  Let  x  and  36  —  a;  =  the  two  parts, 
and  X  (36  —  a;),  or  36  a;  —  a;*  =  m. 
Then,  a;  =i  18  ±  \/^24:  -  m. 

Now,  m  =  324  is  a  maximum ; 

x  =  18  and  36  -  a;  =  18. 


386 


ADVANCEB  ALGEBRA. 


704.  General  Method.— 

Let  mn  he  the  graph  ot  y  =  f(x). 


Y 

/^ 

^^ 

— pVU 

/a 

A 

X 

/ 

/ 

P5^ 

J. 

1 

V 

V     V 

Conceive  a  point,  P,  to  move  along  the  graph,  carrying  with  it  a 
tangent  Hne  to  the  graph,  in  such  a  manner  as  to  cause  the  abscissa 
(ic)  of  the  point  to  increase  uniformly.  Let  v  be  the  value  of  the  vari- 
able angle  which  the  tangent  line  makes  with  the  ic-axis.  At  P' 
1?  <  90 ;  hence,  tan.  v,  or  /i  {x\  is  positive  [668,  Sch.].  This  is  true, 
however  near  Pi  is  to  pii.  At  P°  the  tangent  line  is  parallel  to 
the  rc-axis;  hence,  v  =  0,  and  tan.  v,  or  fx{x)  =0.  At  P™,  t;>  90; 
hence,  tan.  v,  or  /i  (a;),  is  negative  [668,  Sch.].  This  is  true,  however 
near  P™  is  to  P°.  Again,  just  before  P  arrives  at  P^,  v  >  90°,  and 
tan.  V  is  negative ;  when  P  is  at  P^,v  =  0  and  tan.  v  =  0;  when  F 
has  just  passed  F^,  v  <  90  and  tan.  v  is  positive.    Therefore, 

706.  Frin,  1»    /i(a;)  =  0  at  turning  values  of  f(x), 

Prin,  2,    Immediately  before   a   maximum   value  of 

f{x),  f\{x)  is  positive y  and  immediately  after,  negative, 
Prin*  3,    Immediately    before  a  minimum   value  of 

/(^)>  /i(^)  *^  negative,  and  immediately  after,  positive. 

706.  Caution  1.— A  root  of  /i  (x)  =  0  is 

not  necessarily  the  abscissa  of  a  turning  point. 
For  a  tangent  line  to  a  graph  may  be  parallel 
to  the  a;-axis  where  there  is  no  turning  point, 
as  where  two  branches  tangent  to  the  same 
line  coalesce  at  the  point  of  tangency.  (See 
diagram.) 

It  is  only  when  Prin.  2  or  Prin.  3  is  satisfied,  as  well  as  /i  (x)  =  0, 
that  a  turning  point  is  established. 

Caution  2. — There  may  be  turning 
points  under  peculiar  conditions  when 
/i  (x)  4=  0.  For  there  may  be  turning 
points  where  the  tangent  line  to  the 
graph  is  not  parallel  to  the  a;-axis; 
as  where  two  branches  coalesce  and 
cease.    (See  diagram.) 


MAXIMA  AND  MINIMA   OF  FUNCTIONS.        387 

707.  Observations. — 1.  So  long  as  f{x)  remains  continuous,  its 
maxima  and  minima  values  succeed  each  other  alternately. 

2.  If  two  successive  turning  values  of  f{x)  have  the  same  sign, 
the  graph  of  f{x)  between  these  values  can  not  cross  the  a;-axis,  or 
f{x)  4=  0  between  these  values. 

3.  If  two  successive  turning  values  of  f(^)  have  opposite  signs, 
the  graph  of  f{x)  must  cross  the  ic-axis  between  these  values,  or 
f{x)  =  0  somewhere  between  these  values. 

4.  \i  x  =  a  and  x  =  h  render  f{x)  =  0,  and  a^h,  there  must  be 
a  turning  value  of  f{x)  between  x  =  a  and  a;  =  6. 

Example. — Find  the  turning  values  of 

f{x)  =  a;3  -  9  a;2  _^  24  a;  +  16. 

Solution :  f{x)  =  a:3  —  9a;2  +  24a;+16 

/,  {x)  =  3a;«  -  18a;  +  24  =  0; 

or,  /,(a:)  =  a;2  — 6a;  + 8  =  0; 

whence,  a;  =  4  or  2,  critical  values. 

/i(a;-Aa:).     ^^4  ,  =  (4  -  A  a;)*  -  6(4- A  a:)  +  8  = - 
(  A  a;  =  o  f 

/i(a;+  Aa;),     ^^4  >  =  (4  +  A  a;)*  -  6(4  +  A  a;)  +  8  =  + 

■)  A  «  =  o  ) 

.'.    f{x)  is  a  minimum  when  a;  =  4 
But  f{x)x  =  4  =  43  -  9  X  42  +  24  X  4  +  16  =  32. 

.*.    Minimum  value  of  f{x)  =  32 

/i(a;-Aa;),     ^^g  .  =  (2  -  a  a;)»- 6(2  -  A  a;)  +  8  = + 

j  A  a;  =  o  j 
fi(x+  Ax)^     ^^2  I  =  (3  +  A  a;)«  -  6(2  +  A  a;)  +  8  =  - 

I  A  a;  =  o  I 

.*.    f(x)x  =  2  is  a  maximum 
But  f(x)x  =  2  =  2»  -  9  X  22  +  24  X  2  +  16  =  36 

.*.    Maximum  value  of  f(x)  =  36. 
The  value  of  f(x)x  =  a  is  best  obtained  by  synthetic  division,  as  in 
Art.  106. 

EXERCISE     101. 

Find  the  maxima  and  minima  values  of  : 

1.  4:a^-15a^-}-12x-l  5.  x^ -dx^  -  9x-\-6 

2.  23^-21a^-{-36x-20  6.  (a; -  1)* (a;  +  3)^ 

3.  x^-{-6x  +  6  7.  (re  -  af  {x  +  bf 
4.ic2_6a;  +  5                              B.  x^ -3x^ -\-3x-\-H 


388  ADVANCED  ALGEBRA. 

9.  a^-^Qx-6  11.  a^-^3(?-{-U 

\0.  a^  —  Qx  —  h  12.  a;*  +  a;3  +  a;2  — 16 

13.  Show  where  a  line  a  feet  long  must  be  divided  so 
that  the  rectangle  of  the  two  parts  may  be  the  greatest 
possible. 

14.  Find  the  altitude  of  the  maximum  cylinder  that  can 
be  inscribed  in  a  sphere  whose  radius  is  r. 

Suggestion.— Let  B C  =  Xy  BD  =  r  —  x,  and  AB  —  y^ 

then,      2/^  =  (^  +  ^)  (r  —  a;)  =  r*  —  ic*,  and 
/(a;)  =  F=ir2/2  X  2 a;  =  3 ir a; (r^  —  jr*) 
/i(a;)  =  2ira;  X  (-2  a;) 
+  (r3_a;8)x2ir  =  0 

whence,  x  =  -^  ^J~^ 
o 

2 
and,       2/2  _  ^3  _  3.3  _     ,.2 

o 

2        /- 

15.  Find  the  altitude  of  the  maximum  cylinder  that  can 
be  inscribed  in  a  cone  whose  altitude  is  a  and  whose  radius 
is  J. 

16.  Find  the  volume  of  the  maximum  cone  that  can  be 
inscribed  in  a  given  sphere. 

17.  Find  the  area  of  the  maximum  rectangle  that  can 
be  inscribed  in  a  square  whose  side  is  a, 

18.  What  is  the  maximum  convex  surface  of  a  cylinder 
the  sum  of  whose  altitude  and  diameter  is  a  constant  a  ? 

19.  Find  the  altitude  of  the  maximum  cylinder  that 
can  be  inscribed  in  a  right  cone  whose  altitude  is  a  and 
the  radius  of  whose  base  is  J. 

20.  Eequired  the  area  of  the  maximum  rectangle  that 
can  be  inscribed  in  a  given  circle. 

21.  Required  the  greatest  right  triangle  which  can  be 
constructed  upon  a  given  line  as  hypotenuse. 


CHAPTER  XII. 
THEORY    OF    EQUATIOJ^S, 


Introduction. 

708.  Equations  of  the  first  and  second  degree  have 
already  been  treated,  and  need  no  further  attention  here. 

709.  Jerome  Cardan,  an  Italian  mathematician  (1501- 
1576),  published  in  1545  a  method  of  solving  cubic  equa- 
tions, now  known  as  *' Cardan's  Formula."  But,  as  this 
formula  is  not  finally  reducible  when  the  roots  of  an  equa- 
tion are  real  and  unequal,  it  is  not  of  much  practical 
value. 

710.  Eene  Descartes,  a  French  mathematician  (1596- 
1650),  transformed  the  general  bi-quadratic  equation  so  as 
to  make  its  solution  depend  upon  that  of  the  cubic  equa- 
tion ;  but,  as  he  invented  no  new  method  of  solving  the 
latter,  the  same  difficulties  are  encountered  in  the  applica- 
tion of  his  rule  as  are  met  in  Cardan's. 

711.  Nicholas  Henry  Abel,  a  Norwegian  mathematician 
(1802-1829),  demonstrated,  in  1825,  the  impossibility  of  a 
general  solution  of  an  equation  of  a  higher  degree  than 
the  fourth.  Previous  to  that  date  many  such  solutions 
were  attempted. 

712.  The  real  roots  of  numerical  equations  of  any  de- 
gree are,  however,  attainable  through  laws  and  principles 
to  be  developed  in  this  chapter. 


390  ADVANCED  ALGEBRA, 

Normal  Forms. 

713.  Theorem  I. — Every  equation  of  one  unhnown 
quantity  with  real  and  rational  coefficients  can  he  trans- 
formed into  an  equation  of  the  form  of 

Ax^-\-Bx^-'^-{-Cx^-^-i-  ....  +i;  =  0, 
in  which  A  and  all  the  exponents  of  x  are  positive  in- 
tegers, and  each  of  the  remaining  coefficients,  including  L, 
is  either  an  integer  or  zero. 

Note.    L  may  be  regarded  the  coefficient  of  a:P. 

Demonstration. — 1.  If  the  equation  contains  fractional  terms,  it 
may  be  cleared  of  fractions. 

2.  If  there  are  any  terms  in  the  second  member,  they  may  be 
transposed  to  the  first  member. 

3.  All  terms  containing  like  exponents  of  x  may  be  collected  into 
one  term  by  addition. 

4.  If  A  is  negative,  both  members  may  be  divided  by  —  1. 

5.  If  X  contains  negative  exponents,  both  members  may  be  multi- 
plied by  X  with  a  positive  exponent  numerically  equal  to  the  greatest 
negative  exponent. 

6.  If  X  contains  fractional  exponents,  x^  may  be  substituted  for  x, 
in  which  m  is  the  L.  C.  M.  of  the  denominators  of  the  fractional  ex- 
ponents. 

The  roots  of  the  transformed  equation  will  he  the  mth  root  of  the 
roots  of  the  original  equation. 

1.  The  terms  may  now  be  arranged  according  to  the  descending 
powers  of  x. 

714.  The  equation 

A  af-i-Bx''--'  +  (7:r"-2-f  . . . .  +  X  =  0, 
is  known  as  the  first  normal  form  of  an  equation  of  one 
unknown  quantity,  and  will  hereafter  be  represented  by 

Example. — Transform  3x^+-r—S-[-7  x~^  =  -  +  -r- 

x^  o       xt 

into  the  first  normal  form,  and  compare  the  corresponding 

roots  of  the  two  equations. 


NORMAL  FORMS,  391 

Solution:  Given  3a;§  +  -r  —  8  +  7a;— f  =  —  +  -j.  (A) 

xi  ^       xt 

Clear  of  fractions,  9x  +  12  —  24:X^  +  21  x—^  =  4xi  +  9xi  (B) 

Transpose  and  collect  terms, 

9a;  +  21a;-i-28a;i-9a;*  +  12  =  0  (C) 

Multiply  by  xt , 

9a;t  +  21-28a:f-9a;l  +  12a;i  =  0  (D) 

Puta;  =  a;«,      9a;9  +  21  -  282;*  -  9^:^  +  i2a:3  _  q  (E) 

Rearrange  terms, 

9a;9  +  0a;8  +  0a;'  +  0a;«-28a;5-9a:4  +  12a:3  +  0a:2  +  0a:  +  21  =  0        (F) 

The  roots  of  (A)  =  Vof  the  roots  of  (F). 


715.  An  equation  that  contains  all  the  powers  of  x, 
from  the  highest  to  the  lowest,  is  called  a  Complete  Equa- 
tion, An  incomplete  equation  may  be  written  in  the  form 
of  a  complete  equation  by  supplying  the  wanting  terms 
with  coefficients  of  zero. 

Thus,  a;^  —  4a;^  +  2a;  —  5  =  0  may  he  written  2:^  ±  0 ar* 
—  ^7?  ±0x^-\-2x  —  b  =  0. 


716.   Theorem  II, — The  equation   F^  {x)  =  0  ?nay  be 
transformed  into  an  equation  of  the  form  of 

x^-\-p,x''--'  +p,x--'-\-., .  .+^„  =  0, 
in  lohich  the  coefficient  of  a:"  is  unity ^  and  each  of  the 
remaining  coefficients  is  either  an  integer  or  zero. 

Demonstration.— 

Take      Fn{x)  =  Ax""  +  Bxf^-'^  ^Cx!^-'^  ■¥ +  L  =  0 

^  ^  X     Ax""      Bx»-^       Cx»-^ 

Put        X  =  -r,  — r-   +       .„     ,     +   -j^r^T  + +L  =  0 

A      A"         A'*-^         A»-2 
Multiply  by  .4„_i , 

a;«  +  Bx^-^  +  A  Cx'^-^  + +  J.«-^  X  =  0 

Put  pi  for  B,  Pi  iov  AC, p„  for  J.«-» L, 

x"*  +  pi  X^-^  +  Pi  X^-^  + +  Pn=  0 

This  is  the  second  normal  form  of  an  equation  of  one  unknown 
quantity,  and  will  hereafter  be  represented  by  /« {x)  —  0. 


ADVANCED  ALGEBRA. 

717.  Cor.  1, — Each  root  of  /„  {x)  =  0  is  A  times  as 
great  as  the  corresponding  root  of  F^  {x)  —  0. 

718.  Cor,  2, — The  coefficient  of  the  second  term  of 
/„  {x)  is  the  same  as  the  coefficient  of  the  second  term  of 
F^  {x),  and  the  succeeding  coefficients  of  /„  {x)  are  obtained 
ly  multiplying  the  succeeding  coefficients  of  F^ix),  in 
order,  ly  A,  A^ ,  A^, ^"~^. 

Note. — If  terms  are  wanting,  supply  them  with  coeflScients  of  0. 

Example. — Transform  the  equation  Ax^  —  dx^-\-2a^  — 
7  =  0  into  an  equation  of  the  form  of  f(x)  =  0. 

Solution : 

Given      Fix)  =  Aafi -Sx^  +  Ox^  +  2x^  +  Ox-7  =  0, 

thenwm  fix)  =  x^-dx^  +  4:x0a^  +  Px2x^  +  ¥x0x-4^x'7=z0  [718] 

or,  fix)  =  x^-dx^  +  d2x^-  1792  =  0. 

The  roots  of  fix)  =  0  are  4  times  as  great  as  those  of  Fix)  —  0. 

EXERCISE     102. 

Transform  the  following  equations  into  equations  of 
the  form  of  f{x)  =  0.  Compare  the  roots  of  the  trans- 
formed equation  with  the  roots  of  the  original  equation. 

1.  Sa^-\-2x^-da^i-l!x^6  =  0 

2.  2x'  +  4:a^-x^-{-x^-1l  =  0 

3.  4:X^-\-3a^-5x^+llx-l=:0 

4.  Sx^-2x^  +  3xi-2x^-{-4:X^-2  =  0 

5.  x-^  4_  2  ic-^  +  3  x-^  -  x-^  -^x-^-2x-^  +  2  =  0 

6-lx+^x-i+lxi-^lx-^  +  3  =  0 

7.  x^'-dx^^2x^-\-dx-^-\-2  =  0 

8.  |^t+J^t_  1^  +  1  =  0 


DIVISIBILITY  OF  EQUATIONS. 


Divisibility  of  Equations. 

719.  Theorem  III. — If  a  is  a  root  of  F^  (x)  =  0,  then 
X  —  a  is  a  factor  of  F^  (x). 

For,  let  F.  {x)  ^  {x  -  a)  =  F^_,  {x)  +  ^ 

then,       {i^„_i  {x)\{x-a)-\-r  =  F^  {x)  =  0 
but,  x  —  a  =  0,  since  x  =  a. 

r  =  0; 
whence,  F„  (x)  -^  (x  —  a)  =  i^„_i  (x), 

720.  Cor.  1. — If  a  is  an  integral  root  of  F^  {x)  =  0, 
it  is  a  divisor  of  the  absolute  term  of  F^  {x)  [163]. 

721.  Cor.  2. — If  X  —  a  is  a  factor  of  F^  {x),  then  a  is 
a  root  of  F^  {x)  =  0. 

For,        F^{x)  =  {F,_,  (x)}  (a;  -  a)  =  0  ; 
whence,  x  —  a  =  0,  and  x  =  a. 

722.  Cor.  3* — If  x  is  a  factor  of  F^  {x),  then  zero  is  a 
root  of  Fr,{x)-0. 


Number  of  Roots. 

723.  Theorem  IV,    F^  {x)  =  0  has  at  least  one  root. 
The  demonstration  of  this  theorem  may  be  found  in 

special  treatises  on  the  Theory  of  Equations.     It  is  too 
long  and  tedious  to  be  introduced  here. 

724.  Theorem,  V.  F^  {x)  =  0  has  n  roots  and  only  n. 
For,  F^  {x)  —  0  has  at  least  one  root.  [T.  IV.J 
Let                 a  =  one  root  of  F^  {x)  =  0  ; 

then,       F„ (x)  =  { F„_t  (x)]  {x  —  a}  =0  [T.  IILl 

.-.  F,,_^{x)  =  0, 
Let  b  =  one  root  of  i^„_i  (x)  =  0  ;  [T.  IV.] 

then,  F„_^  (x)  =  {F^^s  (x)]  {x-b}=0  [T.  III.] 

.-.  F^_,{x)  =  0. 


394:  ADVANCED  ALGEBRA. 

Now,  as  F^  {x)  =  0  is  of  the  nth.  degree,  and  each  time 
a  root  is  removed  by  division  the  degree  is  lowered  by 
unity,  it  follows  that  n  roots  and  only  n  can  be  removed 
before  F^{x)  reduces  to  an  absolute  factor.  Therefore, 
Fn  {x)  =  0  has  n  roots  and  only  n. 

725.  Car,    F^  (x)  =  0  may  he  written 

A{x  —  a){x  —  b){x  —  c) (x  —  l)  =  Q; 

or  simply  {x  —  a){x  —  b)  (x  —  c) (x  —  l)  =  0,  in  which 

there  are  n  factors  of  the  form  of  x  —  r,  the  second  terms 
of  which  are  the  roots  of  F^  {x)  =  0  with  their  signs 
changed,  and  may  he  positive  or  negative,  fractional  or 
integral,  rational,  irrational,  or  imaginary,  subject  only 
to  restrictive  conditions  explained  hereafter. 


Relation  of  Roots  to  Coefficients. 

726.  OThearem  Vl.—If  F^  (x)  =  0  be  put  in  the  form 
of  x^  +  B^  a;»-i  +  Ci  a;*-^  + . . . .  -|-  Xi  =  0,  5^^  dividing  both 
members  of  the  equation  by  A,  the  coefficient  of  x"",  then  will 

1.  Bx  =  the  sum  of  the  roots  vnth  their  signs  changed. 

2.  Ci=  the  sum  of  the  products  of  the  roots  taken  two 
together. 

3.  Di  =  the  sum  of  the  products  of  the  roots  with  their 
signs  changed,  taken  three  together. 

4'  El  =  the  sum  of  the  products  of  the  roots  taken  four 
together.     And  so  on  to 

5.  Li  =  the  product  of  all  the  roots  with  their  signs 
changed. 

Demonstration :  Let  the  n  roots  of  the  equation  he  a,  b,  c, I; 

then,  Fn  (x)  =  x'*  +  Bi  x^-^  +  Ci  x^-^  + +  Li 

=  {X  -  a){x  -  b){x  -  c) . . . .  (X  -  I)  [725]. 
After  which  the  theorem  is  a  direct  inference  from  the  binomial  for- 
mula [587],  and  the  principle  that  "  changing  the  signs  of  an  even 


IMAGINARY  MOOTS.  395 

number  of  factors  does  not  change  the  sign  of  their  product "  [page 
26,  Ex.  3]. 

Cor. — Changing  the  signs  of  the  alternate  terms  of 
F^  {x)  =  0  changes  the  signs  of  its  roots. 


Imaginary  Roots. 

727.  Theorem  VII, — Imaginary  roots  can  enter  F^  {x) 
=  0  only  in  conjugate  pairs. 

For  in  this  way  only  will  their  sum  and  the  sum  of 
their  products  be  real  [657],  as  they  must  be  [713]. 

728.  Cor,  1, — The  product  of  the  imaginary  roots  of 
Fn  {x)  =  0  is  positive. 

For  the  product  of  each  pair  is  positive. 
Thus,  (a  -\-bi)  {a  '-bi)  =  a^-\-  W. 

729.  Cor,  2,  —  When  all  the  roots  of  F„  {x)  =  0   are 

imaginary  the  absolute  term  is  positive. 

Suggestion. — For  the  equation  is  then  of  an  even  degree. 

730.  Cor,  3,  F^  {x)  =  0  has  at  least  one  real  root  oppo- 
site in  sign  to  the  absolute  term,  when  n  is  odd. 

731.  Cor,  4,  F„  (x)  =  0  has  at  least  two  real  roots, 
one  positive  and  the  other  negative,  if  n  is  even  and  the 
absolute  term  is  negative. 

732.  Car.  5, — The  sign  of  F^  {x)  for  any  real  value  of 
X  depends  on  the  real  roots  of  F^  (x)  =  0. 

For  the  product  of  x  —  {a-\-b  i)  and  x  —  {a  —  bi)  = 
{x  —  aY  +  b^,  a  positive  quantity  ;  and  this  is  true  of  every 
pair  of  factors  containing  conjugate  imaginary  terms.   ' 

733.  Cor,  6, — Every  entire  function  of  x  with  real 
and  rational  coefficients  may  be  divided  into  real  factors 
of  the  first  or  second  degree. 


396  ^^  VANGED  ALGEBRA. 


Fractional  Roots. 

734.  Theorem  VIII, — JVo  root  of  /„  {x)  =  0  can  he  a 
rational  fraction. 

Take  fn(x)  =  x^  +  PiX^-''-{-p2X--^+  ....+jo^  =  0 

[716].     If  possible,  let  a;  =  t-  ,  a  rational  fraction  in  its 

lowest  terms.     Then,  by  substitution, 

Jni-       Jn-1       -1-       Jn-2      "h  •  •  •  •  "f  ^«  "  ^. 

Multiplying  by  5**"^  and  transposing  terms,  we  have 

an  integer,  which  is  impossible. 

Scholium. — From  this  theorem  it  follows  that  the  ra- 
tional fractional  roots  of  F^  {x)  =  0  may  be  obtained  by 
transforming  F^  {x)  =  0  into  /«  (x)  =  0  and  dividing  the 
roots  of  the  latter  equation  by  A,  the  coefficient  of  ic"~^  in 
the  former. 


Relations  of  Roots  to  Signs  of  Equation. 

735.  Theorem.  IX. — If  F„  (x)  =  0  has  no  equal  roots, 
then  Fn  (x)  will  change  sign-if  x  passes  through  a  real  root. 

For,  take  F^{x)  =\x  —  a){x  —  b){x  —  c) {x—l)  =  0 

[7^5]  ;  conceive  x  to  start  with  a  value  less  than  the  least 
root  and  continually  increase  until  it  becomes  greater  than 
the  greatest  root.  At  first,  every  factor  of  F^  (x)  is  nega- 
tive, but,  at  the  instant  it  becomes  greater  than  the  least 
root,  the  sign  of  the  factor  containing  that  root  will  be- 
come plus,  while  the  others  remain  minus  ;  whence,  F^  {x) 
will  change  sign.  It  will,  moreover,  retain  its  new  sign 
until  it  passes  over  the  next  greater  root,  when  it  will 
again  change  sign,  and  so  on. 


RELATIONS  OF  ROOTS  TO  SIGNS  OF  EQUATION.    397 

736.  Cor,  1, — If  for  any  two  assigned  values  of  x, 
Fr,  {x)  has  different  signs,  one,  or,  if  more  than  one,  an 
odd  number  of  roots  of  F^  [x)  =  0  lie  between  these  values, 

737.  Cor.  2, — If  for  any  two  assigned  values  of  x, 
F^  {x)  has  the  same  sign,  either  no  root  or  an  even  number 
of  roots  of  Fr,  {x)  =  0  lie  between  these  values, 

738.  Some  of  the  properties  of  F^  {x)  =  0,  already  dis- 
cussed, are  beautifully  illustrated  by  the  following  graph. 

Y 


1.  It  is  seen  that  y  =  Fn{x)=:0  when  a;  =  1,  2,  3,  and  5.  There- 
fore, these  values  of  x  are  roots  of  i^„  {x)  =  0. 

2.  Immediately  before  a;  =  1,  2/  is  positive,  and  immediately  after 
x  =  l,  y  is  negative ;  immediately  before  x  =  2,  y  is  negative,  and 
immediately  after  x  =  2,  y  is  positive,  etc. ;  illustrating  that  when  x 
passes  over  a  real  root,  Fn  (x)  changes  sign. 

3.  At  a;  =  3  two  values  of  y  become  zero ;  therefore,  two  roots 
become  identical,  or,  in  other  words,  3  is  twice  a  root.  Were  the  abso- 
lute term  of  i^„  {x)  so  changed  as  to  make  y  somewhat  less,  the  a:-axis 
would  cross  the  graph  twice  between  a;  =  2  and  a;  =  4,  once  before 
a:  =  3,  and  once  after,  thus  proving  conclusively  the  duality  of  the 
root  3,  when  y  =  0. 

4.  Immediately  before  a;  =  —  2  and  a;  =  6,  the  graph  approaches 


398  ADVANCED  ALGEBRA. 

the  ic-axis,  but  in  each  case  makes  a  turn  before  reaching  it,  prevent- 
ing, thereby,  equal  roots  or  unequal  real  roots.  These  turns  locate  the 
position  of  imaginary  roots.  The  truth  of  this  statement  becomes 
manifest  when  we  suppose  the  absolute  term  of  Fn  {x)  to  so  change  as 
to  cause  y  to  gradually  decrease,  the  a;-axis  will  gradually  arise  and 
finally  touch  the  graph  at  ic  =  —  2,  thereby  making  two  equal  roots, 
and,  if  y  continues  to  decrease,  the  ic-axis  will  cross  both  branches 
above  the  turn  at  x=  —2,  making  two  unequal  real  roots. 

The  student  will  be  interested  in  observing  the  changes  in  the 
roots  if  the  absolute  term  of  the  equation  so  changes  as  to  cause  the 
a;-axis  to  gradually  move  from  the  position  Aix\  to  the  position  A^x^- 

5.  It  must  not  be  assumed,  however,  that  imaginary  roots  always 
denote  a  turning  point  in  the  graph  of  the  equation.  Such  may  or 
may  not  be  the  case. 

739.  If  any  two  successive  terms  in  a  complete  equa- 
tion have  like  signs,  there  is  a  permanence  of  sign  ;  if 
unlike  signs,  a  variation  of  sign.     Thus,  in  the  equation 

cc«  -  5  a;5  +  8  a;*  +  7  a;3  -  3  rc2  +  2  re  -  5  =  0 
there  are  five  variations  and  one  permanence. 

740.  Theorem  X. — No  complete  equation  has  a  greater 
number  of  positive  roots  than  there  are  variations  of  sign, 
nor  a  greater  numher  of  negative  roots  than  there  are  per- 
manences of  sign. 

Demonstration :  Let  the  following  be  the  successive  signs  of  a  com- 
plete equation : 

+     --     +     +     — 

There  are  here  two  permanences  and  three  variations.  To  intro- 
duce another  positive  root,  the  equation  must  be  multiplied  by  x  —  a. 

The  signs  of  the  product  will  readily  appear  from  the  following 
work : 

+     —     -     +     +     — 
-I-     — 

+     —     —     +     +     — 
—     +     +     --     + 


+     — 


The  double  sign  denotes  a  doubt,  growing  out  of  an  ignorance  of 
the  relative  numerical  magnitudes  of  the  terms  added. 

Now,  a  careful  inspection  will  show  that,  whether  we  regard  both 
doubtful  signs  negative,  both  positive,  or  one  negative  and  the  other 


RELATIONS  OF  ROOTS  TO  SIGNS  OF  EQUATION. 

positive,  the  number  of  permanences  will  not  be  increased,  but  the 
number  of  terms  is  increased  by  one ;  therefore,  the  number  of  varia- 
tions must  be  increased  by  at  least  one.  Since  the  introduction  of  a 
positive  root  introduces  at  least  one  variation,  it  follows  that  the  num- 
ber of  positive  roots  can  not  exceed  the  number  of  variations. 

In  a  similar  manner,  by  introducing  the  factor  x  -\-  a,  it  may  be 
shown  that  the  number  of  negative  roots  can  not  exceed  the  number 
of  permanences  of  sign. 

This  is  Descartes'  celebrated  rule  of  signs. 

741.  Car,  1, — If  all  the  roots  of  an  equation  are  real, 
the  number  of  variations  equals  the  number  of  positive 
roots,  and  the  number  of  permanences  equals  the  number 
of  negative  roots. 

742.  Cm*.  2, — An  equation  whose  terms  are  all  positive 
can  have  no  positive  roots, 

743.  Cor.  3. — An  equation  tvhose  terms  are  alternately 
positive  and  negative  can  have  no  negative  roots. 


Limits  of  Roots. 

744.  A  nnmber  known  to  be  equal  to  or  larger  than 
the  largest  root  of  an  equation  is  called  a  superior  limit 
to  the  roots  of  the  equation. 

745.  A  number  known  to  be  equal  to  or  smaller  than 
the  smallest  root  of  an  equation  is  called  an  inferior  limit 
to  the  roots  of  the  equation. 

746.  Theorem  XI. — If  the  first  h  coefficients  of  F^  (x) 
are  positive,  and  P  is  the  smallest  of  them,  then,  if  Q  is 

numerically  the  largest  subsequent  coefficient,  \/  ^  + 1  is 
a  superior  limit  to  the  roots  of  F^  {x)  =  0. 

Demonstration :  It  is  evident  that  the  case  in  which  x  must  have 
the  greatest  value  to  make  Fn  (x)  =  0  when  the  first  h  coefficients  are 
positive,  is  the  one  in  which  these  coefficients  are  all  equal  to  the  least 
one  of  them  (P),  and  the  remaining  n  +  1  —  h  coefficients  are  all 


400  ADVANCED  ALGEBRA. 

negative  and  each- equal  to  the  greatest  among  them  {Q).    Therefore, 
the  value  of  a;  is  a  superior  limit  to  the  roots  of  Fn  {x)  —  0,  if 

Pa^  +  i  —  Paari  +  i-k  _  Qx'^  +  ^-^—Q 
or,  z  —  z 

X— 1  X— 1 

or,  Pa?'  +  i-A(a;*-l)  =  ^(a;»  +  ^-*-l) 

or  if,  P(x^  -  1)  =  ^,  since  1  >  (l  -  ^^^^^ 

+  1. 


=  l/| 


747.  Cor, — If  the  signs  of  the  alter^iate  terms  of  an 
equation  he  changed,  then  loill  the  superior  limit  to  the 
roots  of  the  transformed  equation,  with  its  sign  changed, 
be  the  inferior  limit  to  the  roots  of  the  original  equation 
[726,  Cor.].  

Equal  Roots. 

748.  Theorem  XII, — If  F^  {x)  =  0  has  equal  roots,  it 
may  he  separated  into  two  or  more  equations  with  unequal 
roots. 

This  is  a  direct  inference  from  Art.  701. 


Commensurable  Roots. 

749.  The  integral  and  rational  fractional  roots  of  F^  (x) 
=  0  are  called  its  commensurable  roots. 

750.  Problem  1.    To  find  the  commensurable  roots  of 

Fn  {X)  =  0. 

Solution :  Pursue  the  following  line  of  investigation : 

1.  Determine  the  number  of  roots  the  equation  has  [724]. 

2.  Determine  how  many  roots  may  be  positive  and  how  many 
negative  [739]. 

3.  Determine  the  limit  to  the  positive  and  the  negative  roots 
[746,  747]. 


COMMENSURABLE  ROOTS.  401 

4.  Determine  what  integral  numbers  may  be  roots  [720]. 

5.  Find  and  remove  the  integral  roots  by  synthetic  division  [719, 
105]. 

6.  Determine  whether  there  are  any  equal  roots  [701],  and  if  so, 
remove  them  by  synthetic  division. 

7.  Find  the  rational  fractional  roots  from  the  equation  resulting 
from  the  removal  of  the  integral  roots,  and  according  to  Theorem 
VIII,  Scholium. 

niustrations. — 1.  Find  the  commensurable  roots  of 
F^  (x)  =  24  cc*  +  122  a;3  +  5  a;2  -  26  a;  -  5  =  0. 

Solation : 

1.  This  equation  has  four  roots,  all  real,  or  two  real  [724,  731]. 

2.  There  are  one  variation  and  three  permanences  of  sign ;  there- 
fore, there  can  not  be  more  than  one  positive  nor  more  than  three 
negative  roots  [739]. 

3.  The  only  integral  roots  possible  are  +1,  —  1,  +5,  and  —  5 
[720]. 

4.  The  largest  positive  root  <  4/  —  +  1,  or  <  2  [746]. 

5.  Neither  +  1  nor  —  1  is  a  root,  since  F^  (x)  is  not  divisible  by 
either  a;  —  1  or  a;  +  1,  as  witness : 

-1)24  +  122+      5-   26-     5 


-  24- 

98  + 

93- 

67 

+ 

98- 

1)24  +  122  + 
24  + 

93+  67- 

5-26- 

146  +  151  + 

72 

5 

125 

146  +  151  +  125  +  120* 
Note. — It  is  evident  that  when  +  1  is  a  root  of  Fn  ix)  =  0,  the 
sum  of  the  positive  coefficients  must  equal  the  sum  of  the  negative 
coefficients ;  and,  if  —  1  is  a  root,  +  1  is  a  root  if  the  signs  of  the 
alternate  terms  are  changed.  These  facts  determine  a  more  expe- 
ditious method  of  testing  whether  either  +  1  or  —  1  is  a  root. 

6.  —  5  is  a  root,  as  witness : 

-5)24  +  122+    5-26-5 
-120-10  +  25  +  5 
+      2-5-1 
7.'  The  resulting  equation,  after  removing  the  root  —  5,  is  F^  (x) 
=  Mx^  +  2 x^  —  5 X  —  1  =  0,  which  has  no  integral  roots. 

Transform  Fa  (x)  =  0  into  an  equation  of  the  form  of  /a  {x)  =  0, 
f3(:x)  =  afi  +  2x^-  120 a:  -  576  =  0  [733,  Sch.]. 


402  ADVANCED  ALGEBRA. 

8.  /s  {x)  =  0  has  three  roots  [724],  only  one  of  which  can  be  posi- 
tive, and  the  largest  positive  root  possible  is  ^^77  =  24. 

9.  The  divisors  of  576  not  exceeding  24  are  1,  2,  3,  4,  6,  8,  9,  12, 
16,  18,  24.  From  the  relative  values  of  the  positive  and  negative  co- 
efficients it  will  be  seen  at  a  glance  that  a;  >  6. 

10.  +  8  and  +  9  are  not  roots,  but  +  12  is  a  root,  as  witness : 

4-8)1+    2-120-576 


+    8  + 

80- 

-320 

+  10- 

40- 

-896* 

+  9)1 

.+    2- 

120- 

-576 

+    9  + 

99- 

-189 

+  11- 

21- 

-765* 

hl2)l 

.+    2- 

120- 

-576 

+  12  +  168  +  576 

14+    48 

11.  The  resulting  equation,  after  removing  the  root  +  12  from 
/a  ix)  =z  0,  is  /a  (a:)  =:  a;2  +  14  a;  +  48  =  0  ;  whose  roots  are  found  to  be 
-  8  and  -  6  [3311. 

12         8 

12.  The  four  roots  of  Fiix)  =  Q  are,  therefore,  —5,  +04.'  "oi* 

and  -  ^  [733,  Sch.],  or  -5,  +  -^,  -  g  ,  and  -  ^. 


2.  Find  the  commensurable  roots  of 

/^  (a;)  =  a;^ -f  3  a;«  -  1 2  a;5  -  3  6  a:* -f  48  a:^  ^ 

144  a;2_g4^_  192  =  0. 

Solution  :  It  may  readily  be  found  by  synthetic  division  that  +  2, 
—  2,  and  —  3  are  the  only  integral  roots  of  this  equation. 

The  resulting  equation,  after  removing  these  roots,  is  fi  (a;)  =  a:*  — 
8  a:*  +  16  =  0.  If  any  of  the  roots  of  this  equation  are  integral,  they 
must  be  equal  to  one  or  more  of  the  roots  already  found.  They  may, 
however,  all  be  incommensurable  or  imaginary. 

Factoring  ft  (a;),  we  have  {x^  —  4)  (a;*  —  4)  =  0 ; 
whence,  a;  =  +  2,  —  2,  +2,  —  2. 

Therefore,  all  the  roots  of  /t  (x)  =  0  are  ±2,  ±2,  ±  2,  and  —  3. 

EXERCISE     103. 

Find  the  commensurable  roots  of  : 
1.  a^-^x^-\-'ix-b  =  ^  2.  a:3_6a;2-f-10a;-8  =  0 

3.  a^  -  11 2:2  -f  41  2;  -  55  =  0 


COMMENSURABLE  ROOTS.  403 

4.  a:3  +  6a:2_j_i4^_{.12  =  0 

5.  :r*-3a;3_2a;2  +  12a:-8  =  0 

6.  12:^3  _|_3a;2_3^._2  =  o 

7.  a:*  +  2a:3-7a;2-8a;  +  12  =  0 

8.  2r*  -  8  a;3  +  10  a;2  +  24  a;  +  5  =  0 

9.  a;5  +  3a;4-3a:3-9a:2-4a;-12  =  0 

10.  a;*-5ic3  +  3a;2  +  2iC  +  8  =  0 

11.  8a:3_i6^8_3^_|_2i  =  o 

12.  16a;^-48a:3_^  32^:2  _^12a;-9  =  0 

13.  3a;5  +  2a:*-21a:3-14a:2  +  36a;  +  24  =  0 

14.  9  a;5  +  81  :r*  +  203  a:3  +  99  a,-2  -  92  a;  -  60  =  0 

15.  18a;5  +  9a;*  +  22a;3  +  lla;2-96a;-48  =  0 

16.  a:*  +  4ic3_i3  2;3_28a;  +  60  =  0 

17.  ar*  +  2a:3_ii^2_12a;  +  36  =  0 

18.  ic5  +  4a;*  +  a:3_iQ^_4^_|.3^0 

19.  3  a;6  +  22  a;5  +  8  a;*  -  42  :z;3  _  Y9  ^2  _  52  ^  _  12  =  0 


Incommensurable  Roots. 

751.  The  incommensuraUe  roots  of  an  equation  are 
best  sought  for  after  all  the  commensurable  roots  have 
been  removed  by  division  and  the  resulting  equation  trans- 
formed into  an  equation  of  the  form  of  /«  {x)  =  0. 

752.  The  first  step  necessary  in  the  search  for  the 
values  of  the  incommensurable  roots  of  an  equation  is  to 
find  the  number  and  situation  of  such  roots. 

Jacques  Charles  Frangois  Sturm,  a  Swiss  mathemati- 
cian (1803-1855),  discovered  a  method  of  doing  this  in 
1829,  known  as  Sturm's  method. 


404  ADVANCED  ALGEBRA. 

753.  Sturm's  Series  of  Functions.  —  Assuming  that 
f^  (x)  =  0  has  no  equal  roots,  this  eminent  mathematician 
formed  a  series  of  functions,  as  follows  : 

The  first  two  terms  of  the  series  are  /„  (x),  and  its  first 
derivative,  which  we  will  now  represent  by  /«_i  (x). 

The  other  functions,  and  which  are  called  Sturmian 
functions,  are  derived  as  follows  :  Divide  /„  (x)  by  /„_i  (x), 
and  represent  the  remainder  with  its  sign  changed  by 
/«_2  {x).  Divide  /„_i  {x)  by  /„_2  {x),  and  represent  the  re- 
mainder with  its  sign  changed  by  /„_3  (x) ;  continue  this 
process  until  the  last  remainder  with  its  sign  changed  is 
an  absolute  term.  Eepresent  this  remainder  /o  {x).  There 
will  then  be  n-\-l  of  these  functions,  as  follows  : 

/n  (^),  /«-l  {X),    fn-2  {x)  ....fo  [x). 
Caution. — Care  must  be  taken  in  the  operation  of  successive  divis- 
ion not  to  reject  any  negative  factors  except  in  the  remainders. 

764.  Relation  of  the  terms  of  Sturm's  series  of  func- 
tions.— If  we  put  qi,  qz,  q^ as  the  successive  quo- 
tients obtained  in  finding  the  Sturmian  functions,  it  is 
evident  that 

/»(^)=/«-l(^)^l-/n-2W  (1) 

/„_!  {X)    =  /._2  (X)  q2    -  fn-3  (^)  (^) 

fn-2  (X)    =  fn-3  (x)  §'3    "  /«-4  (^)  (3) 

/„_3  (x)  =  ./;_4  (x)  q,  -  /„_5  (x)  (4) 

/„_4  (x)  =  /«_5  (x)  q,  -  /„_6  (x)  (5) 

etc.,  etc.,  etc. 

755.  Fundamental   Principles. 

1.  No  two  consecutive  functions  can  vanish,  i.  e.,  be- 
come 0,  for  the  same  value  of  x. 

For,  if  possible,  \Qtx  =  a  make  /„_2  =  0  and  /„_3  =  0 ; 
then   will  /„_4  =  0    [754,   3],    and   hence,    too,   /n_6  =  0 


INCOMMENSURABLE  ROOTS.  405 

[754,  4],  and  so  on  until  lastly  /« (:?;)  =  0 ;   but  /o  {x)  is 
the  absolute  term  and  can  not  be  zero.     Therefore,  etc. 


2,  If  any  one  of  the  functions  intervening  hetween 
/„  {x)  and  /o  {x)  vanishes  for  any  value  of  x,  the  two  ad- 
jacent functions  have  opposite  signs  for  this  value. 

Thus,  \i  x=:a  causes  /„_3  {x)  to  vanish,  /„_2  (x)  — 
-f.-,{x)  [754,  3].         

3.  If  any  value  of  x,  as  x  =  a,  causes  any  intervening 
function  to  vanish,  then  will  the  number  of  variations  and 
the  nuniber  of  permanences  in  the  signs  of  the  functions 
he  the  same  for  the  immediately  preceding  and  the  imme- 
diately succeeding  values  of  x,  i.  e.,  for  x  =  a  —  <z>  and 
x  =  a-\-  <^ . 

For  the  two  adjacent  functions  will  have  opposite  signs 
when  x-=a  [755,  2],  and  will  not  change  their  signs  for 
any  value  of  x  from  x  =  a  —  c>  and  a;  =  «  -|-  o  ,  since  no 
root  of  either  can  lie  between  these  values  [755,  1].  But 
the  function  in  question  does  change  its  sign,  since  x  passes 
over  a  root  of  the  function  in  going  from  a;  =  a  —  o 
to  ^  =  «  +  o .  If  the  signs  of  the  three  functions  for 
x  =  a  —  <^  are  +?  +>  — >  for  x  =  a-\-o  they  will  be 
+  ,  — ,  — ,  which  in  either  case  form  one  permanence 
and  one  variation.  Similarly,  +,  — ,  —  will  change  to 
+  ,+,—;  — ,  +,  +  will  change  to  —,—,+;  and 
— ,  — ,  +  will  change  to  — ,  +,  +. 


4'  If  any  value  of  x  causes  /„  {x)  to  vanish,  then  will 
one  variation  in  the  signs  of  the  functions  be  lost  in  pass- 
ing from  the  immediately  preceding  value  of  x  to  the  im- 
mediately succeeding  value. 


{X  -  «„)  + 

(1st  term) 

(*-««)  + 

(2d  term) 

{x-a,)  + 

(3d  term) 

406  ADVANCED  ALGEBRA. 

Let  f,{x)  = 

{x  —  «i)  {x  -  ttg)  {x  —  a^) (x  —  a,) ; 

then  /,_!  (a;)  = 

{x  —  ttz)  {x  —  as)  {x  —  a^)  . 
(x  —  tti)  {x  —  Us)  (x  —  a^  . 
{x  —  «i)  {x  —  a^  {x  —  a^  . 
{x  —  «i)  {x  —  ttz)  {x  —  a^) (x  —  «„)  +  (4th  term) 

(x  —  tti)  (x  —  ttz)  {x  —  a^) x—a,^_i^      (nth term) 

Now,  if  x  equals,  say  as,  then  will  fn(^)  and  all  the 
terms  of  /„_i  (x),  except  the  third,  vanish. 

Now,  the  third  term  of  /„_i  (x)  contains  all  the  factors 
of  fn  (x)  except  x  —  a^.  Therefore,  if  x  is  infinitesimally 
less  than  %,  x  —  %  will  be  negative  and  /„  (x)  and  /„_i  {x) 
will  have  opposite  signs  or  will  form  a  variation  ;  but,  if  x 
is  infinitesimally  greater  than  as,  x  —  as  will  be  positive 
and  /„  (x)  and  /„_i  (x)  will  have  like  signs  or  will  form  a 
permanence.     Therefore,  a  variation  is  lost  in  passing  from 

tts  —  O    to    «3  -j-  O  . 

766.  These  principles  are  true  if  /„  (x)  contains  imagi- 
nary roots  as  well  as  when  all  the  roots  are  real,  since  the 
signs  of  the  functions  depend  wholly  upon  the  real  factors 
they  contain  [732]. 

Sturm's  Theorem. 

757.  The  number  of  variations  of  sign  lost  in  the  terms 
of  the  Sturmian  series,  as  the  value  of  x  continuously 
changes  from  a  to  b,  a  being  less  than  h,  equals  the  num- 
ber of  real  roots  of  f„  (x)  =  0  lying  between  a  and  b. 

Demonstration. — For  each  time  the  value  of  x,  in  ascending  from 
a  to  b,  passes  over  a  root  of  /» {x)  =  0,  there  is  lost  one  variation  of 
sign  [755,  4]  and  only  one  [755,  3]. 


INCOMMENSURABLE  ROOTS.  407 

758.  Cor,  1. — The  theorem  is  equally  true  for  F„  {x) 
=  0,  there  being  nothing  in  the  demonstration  of  it  to 
restrict  its  application  to  /«  {x)  =  0. 

769.  Cor,  2, — The  difference  between  the  number  of 
variations  when  +  oo  and  —  cc  are  substituted  for  x  in 
the  series  is  the  number  of  real  roots  in  the  equation. 

760.  Cor.  3, — The  difference  between  the  number  of 
variations  when  0  and  +  oo  are  substituted  for  x  is  the 
number  of  positive  roots,  and,  when  0  and  —  oo  are  sub- 
stituted for  X,  the  number  of  negative  roots. 

761.  Eemark  1.— It  is  evident  that  the  sign  of  the  absolute  term 
of  a  function  is  the  sign  of  the  value  of  the  function,  when  a;  =  0. 

762.  Remark  2. — The  sign  of  the  first  term  of  a  function  is  the 
sign  of  the  value  of  the  function,  when  a;  =  ±  oo . 

For,  Ax^  =  Acc^-^  .x>  Bx:^-'^  +  Ca;"-*  +  Dai^-^  + 

+  Lx""-^  >  Bx^-^  +  Cx:^-^  +  Dx?"-^  + +  L,  when  a;  =  ±  oo . 

763.  Remark  3. — The  sign  of  the  value  of  a  function  for  any 
integral  or  decimal  value  of  x  is  best  determined  by  the  method  ex- 
plained in  Art.  106. 

Illustration. — Find  the  sign  of  i^4  (a:)  =  3  a:*  —  2  a;^  + 
7a;2  — 3a;-8  when  x=l'2. 

Solution :  The  value  of  i^4 {x)  when  a:  =  1-2  is  +  -3808,  as  witness: 
1.3)3  _  2     +7-3-8 
3-6  +  1-92  +  9-984  +  8-: 


1-6  +  8-92  +  6-984  +    -3808* 
.'.  The  sign  of  F^{x)  is  +.' 

Note. — In  practice  it  is  usually  not  necessary  to  make  the  last 
multiplication  and  addition  to  determine  the  sign  of  the  value. 

764.  Remark  4. — Though  it  is  not  usually  best  to  apply  Sturm's 
method  of  solution  to  equations  before  the  commensurable  roots  have 
been  removed  by  division,  on  account  of  the  great  labor  involved  in 
deriving  and  evaluating  the  different  functions  when  the  equation  is 
of  a  high  degree,  yet  such  a  course  may  be  pursued.  If  there  are 
equal  roots,  the  fact  will  appear  in  deriving  the  functions,  and  if  there 
are  integral  or  fractional  roots  they  will  be  discovered  in  evaluating 
the  functions  to  determine  their  signs. 


408  ADVANCJSn  ALGEBRA. 

Example. — Determine  the  number  and  situation  of  the 
real  roots  in  /s  (a:)  =  o^  —  12  a;^  +  57  a;  —  94  =  0. 

Solution :  /a  {x)  =  a;^  — 13  a;2  +  57  a;  -  94 

/2(a;)  =  3a;2-24a;  +  57 

fi{x)  =  —x  +  ^ 

foix)=- 

Substituting  in  these  functions  as  follows,  we  shall  have : 

For  x=+Go,  +     +     —     —        one  variation. 

For  X  =  0,  _     4-     4.     _        two  variations. 

For  x=  —cOy  _     +     +     _        two  variations. 

There  is,  therefore,  one  real  root  between  0  and  4-  oo .  There  is 
no  negative  root.    Therefore,  there  are  two  imaginary  roots. 

To  find  the  situation  of  the  real  root,  we  proceed  as  follows : 
For  re  =  1,  we  have        —     +     +     _        two  variations. 
For  a;  =  2,  we  have        —     +     +     _        two  variations. 
For  ic  =  3,  we  have        —     +     ±     _        two  variations. 
For  aj  =  4,  we  have        +     +     —     —        one  variation. 

Therefore,  there  is  one  real  root  between  3  and  4,  or  the  first 
figure  of  the  real  root  is  3. 

To  find  the  next  figure,  we  proceed  as  follows : 
For  X  —  3-1,  we  have     —     +     _     —        two  variations. 
For  a;  =  3*2,  we  have     —     +     _     _        two  variations. 
For  X  =  3-3,  we  have     —     +     _     _        two  variations. 
For  X  =  3-4,  we  have      +     +     —     —        one  variation. 

Therefore,  the  root  lies  between  3*3  and  3'4,  or  the  first  two  fig- 
ures of  the  root  are  3-3. 

By  a  continuation  of  this  process  the  root  might  be  extended  to 
any  number  of  figures.  A  more  expeditious  method,  however,  is 
known,  and  will  be  explained  hereafter,  for  extending  a  root  after  a 
sufficient  number  of  figures  have  been  found  to  distinguish  the  root 
from  any  other  root  lying  near  it.  Thus,  if  an  equation  had  the  two 
roots  3*1256. .  and  3*1234. . .,  the  first  four  figures  of  each  root  only 
would  be  found  by  Sturm's  theorem. 

When  it  is  known,  as  in  the  above  example,  that  only  one  real 
root  lies  between  two  numbers,  it  becomes  necessary  only  to  study 
the  signs  of  /„  (x),  since  passing  over  the  roots  of  the  intermediate 
functions  does  not  cause  a  change  in  the  number  of  variations. 

The  same  conclusion  will  be  reached  by  the  simple  application  of 
Art.  735,  since  /a  (x)  changes  sign  between  a;  =  3  and  a;  =  4. 


INCOMMENSURABLE  ROOTS.  409 

EXERCISE     104. 

Find  the  number  and  situation  of  the  real  roots  in  the 
following  equations  : 

1.  a:3_4a;2-6a;  +  8  =  0       4.  x^-10a^-[-Qx-\-l  =  0 

2.  x^  +  6x^'-Sx  +  9  =  0       5.  2:r*-lla;2^82,_16_0 

3.  a^-{-3s^-'6x-[-2  =  0       6.  xf"  -  Ux^  +  Ux -3  =  0 

Horner's  Method  of  Root  Extension. 

765.  In  1819,  W.  G.  Horner,  an  English  mathema- 
tician, published  an  elegant  method  of  extending  a  root 
of  an  equation  to  any  desired  number  of  places,  after  a 
sufiBcient  number  of  initial  figures  have  been  found  by- 
other  methods  to  distinguish  the  root  from  other  roots  of 
the  equation.  This  method  is  based  upon  the  following 
principle  : 

766.  Principle, — If  F^  {x)  he  continuously  divided  hy 
X  —  a,  the  successive  remainders  will  he  the  coefficients  in 
inverse  order  of  an  equation  whose  roots  are  a  less  than 
the  roots  F^  {x)  =  0. 

Demonstration : 

Take      En  (x)  =  A  a?— ^  +  5a:«-2  + 

+  Jx^  +  Kx  +  L  =  0  (A) 

Put       Xi  +  a  =  X,  or  Xi=x  —  a; 

En  {Xi  +  a)  =  A(xi  +  «)«-!  +  i?  (a;,  +  a)«-2  + 

+  J{Xi  +  af  4-  K{xi  +  a)  +  i  =  0  (B) 

Expand  terms,  bracket  coefBcients  of  like  powers  of  Xx ,  and  rep- 
resent the  coefficients  of  the  transformed  equation  by  -4j ,  ^i ,  . . . . 
«/i ,  Kx,  Li',  then, 

En  {Xx)  =  Ax  a;i»-i  +  Bx  Xx^-^  + . . . . 

Jx  Xx^  -{■  KxXx-\-  Lx  =  0  (C) 

Now,  the  roots  of  (C)  are  evidently  a  less  than  those  of  (A). 
Substitute  Xi  =  x  —  a  m.  (C), 

En{x -a)  =  Ax{x-  ay  +  Bx(x-  a)"-!  + 

+  Ji{x-  af  +  Kx{x-a)  +  Lx  =  0  (D) 

Of  1  HE 


-NIVERSITY  \ 


410  ADVANCED  ALGEBRA. 

Now,  Fn  (x  —  a)  is  evidently  equivalent  to  Fn  (x),  and  will  leave 
the  same  remainder  when  divided  by  a;  —  a  as  will  Fn  (x)  -i-{x  —  a). 
But,  if  Fn  {x  —  a)  is  continuously  divided  by  x  —  a,  the  successive 

remainders  will  he  Li,  Ki,  Ji, £i,  and  Ai,  or  the  coefficients 

of  Fn  (xi)  in  inverse  order.    Therefore  the  theorem. 

Applications. 

1.  Transform    aH^ -\- x^ -}- x^ -^  3  x  —  100  =  0    into    an 

equation  whose  roots  are  2  less  than  those  of  the  given 

equation. 

Foniii 

(  +  3 


Famii 

+  1 

+    1 

+    3 

-100 

+  2 

+    6 

+  14 

+    34 

+  3 

+    7 

+  17 

-    66* 

+  2 

+  10 

+  34 

+  5 

+  17 

+  51* 

+  2 

+  14 

+  7 

+  31* 

+  2 

The  transformed  equation  is  x^  +  9  x^  +  31  x^  +  51x  —  66  =  0. 

Explanation. — Dividing  by  x  —  2,  by  synthetic  division,  the  co- 
efficients of  the  first  quotient  are  1  +  3  +  7  +  17,  and  the  first  re- 
mainder is  —  66,  the  absolute  term  of  the  transformed  equation. 

Dividing  1  +  3  +  7  +  17  again  by  +  2,  the  second  quotient  is 
1  +  5  +  17,  and  the  second  remainder,  or  the  coefficient  of  x  in  the 
transformed  equation,  is  +  51. 

Dividing  1  +  5  +  17  again  by  +  2,  the  third  quotient  is  1  +  7, 
and  the  third  remainder,  or  the  coefficient  of  x^  in  the  transformed 
equation,  is  +  31. 

Dividing  1  +  7  again  by  +  2,  the  fourth  quotient  is  1,  and  the 
fourth  remainder,  or  the  coefficient  of  a;'  in  the  transformed  equation, 
is  +9. 

Therefore,  the  transformed  equation  is  x*  +  9x^  +  dlx^  +  51x  — 
66=0. 

Query. — Could  you  tell  by  inspection  that  the  roots  of  the  trans- 
formed equation  are  less  than  those  of  the  original  equation  ? 

Query.— Since  1  +  9  +  31  +  51  >  66,  can  a;*  ^.  9  ^^a  +  31  ^s  +  51  a; 
—  66  =  0  have  a  positive  root  equal  to  or  greater  than  unity  ?  Why, 
or  why  not  f 


INCOMMENSURABLE  ROOTS.  411 

2.  Transform  rz;*  +  9  a;^  +  31  a;^  +  51  a;  -  66  =  0  into  an 
equation  whose  roots  are  '8  less  than  those  of  the  given 
equation. 

1+9  +31  +51  -66  (-8 

-8  7-84  31-072  656576 

-0-3424* 


9-8 
-8 

38-84 
8-48 

82-072 
37-856 

10-6 

-8 

47-32 
9-12 

+  119-928* 

11-4 

-8 

+  56-44^ 

12-2* 
The  coefficients  of  the  first  quotient  are  1  +  9-8  +  38-84  +  82-072, 
and  the  first  remainder  is  —  0-3424,  which  is  the  absolute  term  of  the 
transformed  equation.  The  second  remainder,  or  the  coefficient  of  rr, 
is  119-928.  The  third  remainder,  or  the  coefficient  of  x^,  is  56-44. 
The  fourth  remainder,  or  the  coefficient  of  ofi,  is  12-2. 
The  transformed  equation  is 

Q^  +  12-2  a;3  +  56-44  a;^  +  119-928  a:  -  -3424  =  0. 

3.  Transform 

ir*  +  12-2 0^3  _|_  56-44 a;2  +  119*928 a;  -  -3424  =  0 
into  an  equation  whose  roots  are  '002  less  than  those  of 
the  given  equation. 

1        12-2  56-44  119-928  --3434 (-003 

•002  -024404  -112928808       + -240081857616 


12-202  56-464404  120-040928808      -'102318142384* 

•002  -024408           -112977624 

12-204  56-488812  120-153906432* 

•002  -024412 


12-206      56-513224* 
•003 


12-208* 
The  coefficients  of  the  first  quotient  are  1  +  12-202  +  56-464404  + 
120-040928808,  and  the  first  remainder,  or  the  absolute  term  of  the  trans- 
formed equation,  is  —  -102318142384.  The  second,  third,  and  fourth 
remainders,  or  the  coefficients  of  x,  x^,  and  x^,  are  120-153906432, 
56-513224,  and  12-208. 

The  transformed  equation  is 
ic*  +  12-208x3  +  56-513224  a;2  +  120-153906432  a;  -  -102318142384  =  0. 


412 


ADVANCED  ALGEBRA. 


4.  The  integral  part  of  one  of  the  roots  of  a^  -f  x^  +  ^^ 
+  3  a;  —  100  =  0  is  2.     Extend  the  root. 

Form. 


1  +1 

+  2 
+  3 

+  1 
+  6 

+  7 
+  10 

+  3 

+  14           H 

+  17 
+  34 

+  51* 

-  100      .  1 2-802 

-  34 

-  66* 

+  2 

65-6576 

+  5 

+  17 
+  14 

+  31* 

-0-3424* 

+  2 

31-072 

82-072 
37-856 

119-928* 

+  7 

+  2 
+  9* 

7-84 

.   38-84 
8-48 

47-32 
9-12 

56-44  * 

•240081857616 
- -102318142384* 

•8 

•112928808 

120-040928808 
•112977624 

120-153906432  * 

+  9-8 
•8 

10-6 

•8 

•024404 

56-464404 

•024408 

56-488812 
•024412 

56-513224* 

11-4 

•8 

12-2* 

•002 
12-202 

•002 

12-204 

•002 

12-206 
•002 

12-208* 

Explanation. — 1.  Transform  the  equation  into  one  whose  roots  are 
less  by  2.  The  new  equation  is  ocf^  +  Q  x^  -{-  ^Ix"^  -^^  bl  x  —  QQ  =  0. 
The  roots  corresponding  to  the  one  we  are  considering  will  now  be  a 
decimal. 

2.  Since  (-1)2  =  -01 ;  (-1)3  =  -001 ;  and  (-l)*  =  •OOOl,  the  first  three 
terms  are  small  in  comparison  to  51a:;  therefore,  51  ic  =  66  nearly, 
whence  51  may  be  taken  as  a  trial  divisor  to  find  the  next  figure  of 
the  root,  considerable  allowance  being  made  for  the  omitted  terms. 
At  first  we  would  be  tempted  to  try  "9  for  the  value  of  x.  But,  upon 
transforming  the  equation  into  one  whose  roots  are  less  by  -9,  we  shall 
find  that  the  absolute  term  will  become  positive,  which  shows  that  "9 


INCOMMENSURABLE  ROOTS, 


418 


is  a  superior  limit.  We  therefore  use  -8  for  the  next  term  of  the 
root,  and  transform  the  equation  into  one  whose  roots  are  '8  less.  The 
transformed  equation  is  x^  +  12'2a;3  +  56-44  a;^  +  119-9282;  -  -3424  =  0. 
The  root  of  this  equation  is  now  less  than  -1. 

3.  Omitting  the  first  three  terms  of  the  equation  on  account  of 
their  smallness,  and  using  the  coefficient  of  x  as  a  trial  divisor,  we  see 
that  the  root  is  less  than  '01  and  is  about  -002.  The  next  figure  of 
the  root  is  therefore  0,  and  the  following  one  2.  Transform  the  equa- 
tion into  one  whose  roots  are  less  by  '002 ;  the  resulting  equation  is 
Q^  +  12-208  a;3  +  56-513224  a;^  +  120-153906432  a;  -  •102318142384  =  0. 

The  work  may  be  extended  as  far  as  we  please. 

76  7o  Bemark  1. — When  the  number  of  decimal  places  in  the 
absolute  term  becomes  equal  to  the  number  of  such  places  desired  in 
the  root,  we  may  begin  to  drop  one  figure  in  the  preceding  term  (trial 
divisor),  two  in  the  next  preceding  term,  and  so  on  toward  the  left. 
When  all  the  figures  of  the  first  term  are  exhausted,  the  remaining  fig- 
ures of  the  root  may  be  found  by  simply  dividing  by  the  trial  divisor. 

768.  Bemark  2.— The  absolute  term  after  each  transformation 
must  be  negative,  else  would  the  last  figure  of  the  root  used  be  too 
large  (a  superior  limit). 

769.  Bemark  3. — The  method  may  be  applied  with  equal  facility 
to  extending  an  integral  root  after  a  sufficient  number  of  initial  fig- 
ures have  been  obtained  by  trial  or  by  Sturm's  Theorem  to  distinguish 
the  root  from  others  of  the  equation.  It  may  be  used  with  exactness 
whenever  there  is  an  exact  root ;  hence,  the  incorrectness  of  the  title 
"Horner's  Method  of  Approximation"  given  the  method  by  most 
authors. 

770.  Bemark  4. — The  negative  roots  are  the  numerical  equiva- 
lents of  the  positive  roots  of  the  equation  resulting  from  changing  the 
signs  of  the  alternate  terms,  and  may  be  found  accordingly. 

5.  Solve  a?  -  1728  =  0,  or  find  the  Vi728, 

Solution. 

(12 


+  0 
10 

10 

+  0 
100 

100 
200 

300* 

-1728 
1000 

-728* 

10 
20 

+  728 
0 

10 
30* 

64 
"364 

32 


414 


ADVANCED  ALGEBRA. 


6.  Extract  the  5tli  root  of  4312345   to  thousandths, 
e.,  solve  approximately  x^  —  4312345  =  0. 


Solution. 

1    0 

0 

0 

0       -4312345  1 21-229 

20 

400 

8000 

160000       3200000 

20 

400 

800 

8000 
24000 
82000 
48000 
80000  * 

160000 

-1112845* 

20 

640000 

40 

1200 
1200 

800000  * 

20 

84101 
884101 

884101 
-  228244  * 

60 

2400 
1600 
4000* 

20 

4101 

88304 
972405  * 

80 

84101 
4203 

88304 
4806 

198220-84882 

20 

101 

100* 

-  30028-15168  * 

4101 
102 

18699-2416 
991104-2416 

1 

101 

4208 

103 

4306 

92610* 

18877-3264 

1 

886-208 

1009981-5680  * 

102 

98496-208 

1 
103 

104 
4410* 

890-424 

94886-682 

894-648 

1 

21-04 

104 
1 

4481-04 
21-08 

95281-280* 

105* 

4452-12 
21-12 

•2 

105-2 
•2 

4473-24 
21-16 

105-4 

4494-40  * 

-2 

105-6 

•2 

105-8 

•2 

106-0* 

The  number  of  decimal  places  in  the  second  remainder  is  greater 
than  the  number  required  in  the  root;  therefore,  the  remaining  fig- 
ures may  be  found  by  dividing  the  remainder  by  1009981-568  [767, 
Rem.  1]. 


INCOMMENSURABLE  BOOTS.  4-15 

EXERCISE     108. 

Solve  : 
1.  x^  -  704  a;  -  58425  =  0  2.  a?  -  15348907  =  0 

3.  x^-\-Za?'-^x-'i  =  0 

4.  iC*-4a;3-6a:2  +  32a;-26  =  0 

5.  ir*-19a;3  +  24a;2  +  712a;-40  =  0 

6.  2^5  +  12 a;*  +  59a;3  +  150a:2  +  201a;  +  94  =  0 

7.  3a;*  +  24a:3  +  68a:3_^g2^_964  =  0 

8.  Find  the  cube  root  of  2         9.  Find  the  fifth  root  of  5 

10.  a:3  4-ii<?;2_io2a;+181  =  0 

11.  a:*  +  9ar^  +  31a;2  +  51rr-66  =  0 

12.  a;5  +  2  2;*  +  3  i?^  +  4ic3  +  5  a;  -  54321  =  0 

13.  One  root  of  the  equation  o? -\-2x^ -\-^x  —  13089030  is 

235.     Find  a  cubic  equation  whose  root  is  225. 


Cubic  Equations. 

771.  A  cubic  equation  containing  an  integral  root  may 
be  readily  factored. 

Let  —  a  be  a  root  of  a  cubic  equation,  then  x-\-a  \^ 
a  factor  of  the  equation  [719].  Let  x^-{-mx-\-n  be  the 
other  factor ;  then, 

(x-\-a){ci^-\-mx-{'n)  =  0  (A) 

or,     x^  -\- {a -\- m)  7?  -\- {a  m -\- n)  X -\- a  n  =  0  (B) 

We  now  observe  that  if  we  subtract  the  factor  a  of  the 
absolute  term  from  the  coefficient  of  x^,  and  the  factor  n 
from  the  coefficient  of  .t,  the  latter  remainder  divided  by 
the  former  will  give  a,  the  root  with  the  sign  changed. 
This,  then,  is  the  condition  under  which  a  factor  of  the 
absolute  term  is  a  root  with  the  sign  changed. 


416  ADVANCED  ALGEBRA. 

Illustration. — Solve  a?  —  x^  —  4:X-\-4:  =  0, 

Solution :  1.  The  factors  of  +  4  are  +3  and  +  2 ;  —  2  and  —  2 ; 
+  4  and  +  1 ;  and  —  4  and  —  1 

Try  whether  +  4  is  a  root  with  the  sign  changed. 
Take  —  1  and  —  4,  the  coeflScients  of  x^  and  x. 

Subtract,     +  4  and  +  1,  the  factors  of  the  absolute  term. 
Divide,        —5)         —  5  ( which  =}=  4. 

.'.    4  is  not  a  root  with  the  sign  changed. 
2.  Try  whether  —  2  is  a  root. 
Take  -  1  and  -  4 

Subtract,     +  2  and  +  2 

Divide,        -3)         -6(  +  2 
.'.     —  2  is  a  root. 

Now,  x^  —  x^  —  4tx  +  4:  =  {x  +  2){x^  —  ^x  ■\-  2)  =  0; 
whence,  a;  =  —  2,  2,  and  1. 


Cardan's  Formula. 

772.   I.  The  general  cubic  equation  a^ -\- a  oi? -{- h  x -\- c 
=  0  may  be  transformed  into  an  equation  of  the  form  of 

y''-\-py^q  =  0,  by  putting  x  =  ij-  -a. 

Demonstration :  Take  a^  +  ax"^  +  hx  +  c  =  Q.  (A) 

Put 


x  =  y- 

g-a;  then. 

a^  =  y^- 

-ay^-\-  -a^y-—a^ 

ix^  = 

2              1 

ayi--a^y+—a^ 

hx- 

by   -g-ttft 

c  = 

c 

.'.  a^  +  ax^  +  bx  +  c  =  y^  +  (*- -g-^M^  +  (27^' "s""*  +  ^) 

1  2  1 

Put  p  tor  b—  -^ a*,  and  q  tor  ^a^  —  -^ah  +  c;  then, 

a:^  +  ax''  +  bx  +  c  =  y^+py  +  q  =  0.  (A) 


CUBIC  EQUATIONS.  417 

773.   II.  The  equation  y^-{-py-\-q=^0  may  be  trans- 


formed into  a  quadratic  by  putting  y  —  z—  — 

o  z 


Demonstration:  Take  y^  -^-py  +  { 

1=0. 

Put 

y  =  z- 

P  . 
~  3z' 

then. 

yZ^2% 

P' 

p^ 

27  £3 

py  = 

pz 

P' 
dz 

,*, 

y^  +py  +  q  =  z^ 

— 

pz 
27  z^ 

+  Q  = 

0 

whence, 

27z'^ +  27  qz^- 

pz 

=  0; 

or, 

2«  +      qz^-^ 

^1^' 

=  0. 

(B) 


(C) 


774.   III.  The  roots  of  z^ -\- q  z^  —  ^;;:  p^  =  0  are 

/c7 


p 


-(-l^\/j  +  ff'^'''^''' 


^  3z 


(-f+vT^+(-iVS+#    <»' 

This  is  Cardan's  formula.  ^ 

The  values  of  x  may  be  obtained  by  subtracting  —a 
from  the  values  of  y. 

775.   IV.  Cardan's  formula  fails  when  all  the  roots  are 
real  and  unequal. 

For,  let  the  roots  be  0^  +  V^,  a  —  V~b,  and  c.     Then, 
since  the  coefiBcient  of  y^  is  0, 

(^  a^  ^/b)  -^  (^  a-j-  VJ)  -  c  =  0  [726,  1]  ; 
whence,  c  =  —  2  «.  _ 

The  equation  whose  roots  are  a-j-V^,  a—Vb,  and 
-2a,  is  y^-{Sa''-\-b)y-2{a^-ab)  =  0. 
.-.  p  =  -{da^-\-b),  md  q  =  2{a^-ah), 


whence,   A/^  +  |!  =  L2  _  ^  j  j  V-  3  &,  which 


is  im- 


418  ADVANCED  ALGEBRA. 

aginary,  and,  therefore,  irreducible  when  h  is  positive,  or 
when  the  roots  are  all  real  and  unequal. 

Illustrations.— 1.  Solve  ic^  —  3  a;^  -}-  4  =  0. 

Here  a  =  —  3 ;  hence,  x  =  y  —  I  —^-  j  =  ^  +  1. 

Substitute,  (2/  +  l)^  -  3  (y  +  1)^  +  4  =  0. 

Reduce,  2/3  —  3^  +  2  =  0.  (1) 

P  1 

Here  «  =  —  3 ;  hence,  y  =  z  —  :^  =  z  +  — . 

oz  z 

Substitute,    (^0  +  ^V  -  3  T^  +  i)  +  2  =  0. 

Reduce,  z^  +  2  z^  =  —  1. 

Complete  the  square,  z^  +  2z^  +  1  =  0. 

Extract  the  V*  z^  +  1  =  0. 

Factor,  (z  +  l)iz^  - z  +  1)  =  0 ; 

1        1      /— T 
whence,  z  =  —  i  or-^  i-^V  —  3* 

x  =  y  +  1  =  z  +  —  +1=  —  1,  or  2,  or  2. 


776.  Sometimes  an  integral  root  can  only  he  approxi- 
mately found. 

2.  Solve  x^-\-dx^-{-9x^l3  =  0.  (A) 

Here  a  =  3 ;  hence,  x  =  y  —  l^)=y  —  l. 

Substitute, 

(2/ -  1  f  +  3  (2/ -  1)2  +  9  (2/ -  1)  -  1 3  =  0. 

Reduce,  y^  +  6y-20  =  0.  (1) 

p  2 

Here  p  =  Q;  hence,  y  =  z  —  ^  z=  z . 

^        '  '  ^  dz  z 

Substitute  in  (1), 

(._|)%e(._|)_.o  =  o. 

Reduce,  z^-20z^  =  8.  (2) 

Complete  the  square,  2«  -20z^  +  100  =  108. 
Extract  the  \/,  z^  -  10  =  ±  10-392304. 

Transpose,  ^«  =  20-392304  or  -  -392304+ 

Extract  the  V»  '^  =  2*73+  or  -  -73  + 

x=:y-l  =  z--  r\-l  =  2-73  --73  +  1  =  3,  or  --73  +  2-73  +  1  =  3. 

z 

These  two  values  are  identical.    The  other  two  roots  are  found  by 
dividing  equation  (A)  by  x  —  d. 


CUBIC  EQUATIONS.  419 

EXERCISE     106. 

Solve  : 

1.  a;3-3a:2  +  7a;-5  =  0  8.  iC^H- a;^  _  8a;  —  12  =  0 

2.  x^  —  Q7^-\-l()x  —  S  =  0  9.  a:3_a;2  — 8a;+12  =  0 

3.  rc3  -  11  0:2  +  41 2;  —  55  =  0  10.  ar^  -  11 2;  -  20  =  0 

4.  a:3  +  6a:2^i4^_^12  =  0  11.  a:^  -  26^:+ 60  =  0 

5.  a:3_4a;8  +  5a;-6  =  0  12.  a:^  _  4  ^jS  _|.  3  _.  q 

6.  a?-\-^x'-\-Qx-\-S  =  0  13.  a;3_4a;2_Ya;_^10  =  0 

7.  a:3_|.7^_j_i6a;4-12  =  0  14.  a:3_|_4a;2_  7^;- 10  =  0 


Recurring  Equations. 

777.  A  Recurring  Equation  is  one  in  which  the  co- 
efficients of  the  first  and  last  terms,  and  of  those  equi- 
distant from  the  first  and  last  terms,  are  numerically 
equal,  and  the  signs  of  the  corresponding  terms  are  either 
alike  throughout  or  unlike  throughout ;  as, 

1.  a;5-4a;*  +  5a:3  +  5a:2~4ic  +  l=:0. 

2.  a;^  +  3a:*-2a;3  +  2ic2_3^_l_0. 

3.  a;«  +  4a;5-5a,-*  +  3ic3-5a;2  +  4ir  +  l  =  0. 

778.  In  a  recurring  equation  of  an  even  degree  in 
which  the  corresponding  terms  have  unlike  signs,  the 
middle  term  is  wanting. 

For,  according  to  definition,  it  is  both  positive  and 
negative. 

779.  A  Reciprocal  Equation  is  one  such  that,  if  a  is  a 

root,  —  is  also  a  root. 
a 

780.  Thetyrem  I, — A  recurring  equation  is  also  a  re- 
ciprocal equation. 


+  ;;^  +  ;;^+----±;72±-:;±l  =  o         (C) 


420  ADVANCED  ALGEBRA. 

Demonstration :  Let  a  be  a  root  of 
f^{x)  =  x^  +  Ax^-'^  +  Bx^-'^  + . . .  .±  Bx^  ±  Ax  ±  1  =  0  (A) 

then,       a«  + J.a«-»  +  J5a«-2  + ±Ba^  ±Aa±\  =  0  (B) 

Substitute  —  for  x  in  /„  {x)  =  0, 

whence,  1  +  Aa  +  Ba^+ ±B w^-^ ±  A a»-'  ±  a*  =  0  (D) 

or,  a"  + ^a«-i  +  5a«-2+ ±  ^a^  j|.  ;ia^  ^  1  _  0  (E) 

Now,  (E)  is  identical  with  (B) ;  therefore,  if  a  is  a  root  of  (A), 

—  is  also  a  root.  

a 

781.  Theorem  II. — A  recurring  equation  of  an  odd 
degree  has  -\- 1  for  a  root  when  the  signs  of  the  correspond- 
ing terms  are  unlike. 

Demonstration : 

Let      a;2»  +  i +  ^a;2«  +  ^a;2«-»  + -Bx^-Ax-1=0'    (A) 

then,  (a;2»+i  -  1)  +  Axix^''-^  -  1)  +  Bx^ix^''-^  -  1)  + =0    (B) 

Now,  each  term  of  (B)  is  divisible  hj  x  —  1  [134,  P.]  ; 
.'.    a;  —  1  =  0,  or  x=l. 


782.  Theorem  III, — A  recurring  equation  of  an  odd 
degree  has  —  1  for  a  root  when  the  signs  of  the  correspond- 
ing terms  are  alike. 

Demonstration : 

Let      ir2«  +  i +  J.a;2«  + 5a;2«-i +....+ 5a;2  +  Aaj  + 1  =  0    (A) 

then,  (a;2«  +  i  +  1)  +  Ax(x^''-'^  +  1)  +  Bx^x^-""-^  +  1)  + =0    (B) 

Now,  each  term  of  (B)  is  divisible  by  a;  +  1  [135,  P.]  ; 
.*.    a;  +  1  =  0,  or  a;  =  —  1. 


783.  Theorem,  IV. — A  recurring  equation  of  an  even 
degree  has  + 1  and  —  1  for  roots  when  4he  signs  of  the 
corresponding  terms  are  unlike. 

Demonstration : 

Let      a;2»  +  ^a:'"- ^  +  5a:2«-2+ —  Bx^  —  Ax  —  \  =  (i    (A) 

then,      (x2 «- 1)  +  ^  a:  (a;2«-2  -  1)  +  5  a;2(a;2  "-4-1)+ =0    (B) 

Now,  (B)  is  divisible  by  both  a;  -  1  and  a;  +  1  [134,  136,  P.]  ; 
.*.    a;  —  1  =  0  and  x  +  1  =  0;  whence,  a;  =  ±  1. 


RECURRING  EQUATIONS,  421 

784.  Thetyrem  V, — A  recurring  equation  of  an  even 
degree  may  le  transformed  into  an  equation  of  one  half 
the  degree  when  the  signs  of  the  corresponding  terms  are 
alike. 

Demonstration : 

Let     a;2»+^a;2»»-J +  5a;2«-2+ +  ^a;^  +  ^a;  +  1  =  0     (A) 

Divide  by  x*,  and  collect  terms, 

(^»+l)  +  A(^-.+  ji^)+£(x«-«+jjL)  +  ....+P=0     (B) 

Put       X  ^ =  z  ;  then  will 

X  ' 

x^  +  -.  =  z^-2 
x^ 

and,  in  general,  each  term  of  (B)  may  be  transformed  into  a  term  of 
only  half  the  degree. 


Illustration. — 

Take  a;^  +  ^o;''^  -  3 a:*  +  2 a;^  -  3  2;3  +  4a;  +  1  =  0.       (A) 

Divide  by  a:8,       jc«  +  4a:2-3a;  +  2- -  +  -^  +  -^  =  0  (B) 

•'  X       x^      x^  ^  ' 

Rearrange  the  terms  and  factor, 

Put  x->r  -  =  y\  (1) 

then,  a;2  +  2  +  -2  =  2/^ ;  (2) 

or,  a;2  +  |,  =  2/2-2; 

JO 

3        1 

and,      a;3  +  3a;+ -  +  ^  =  2/«; 


^'^'+^+K'^-'^)=^' 


a;3+  J-=y3_3y.  (3) 


Substitute  (1),  (2),  and  (3)  in  (C), 

(2/3  -  32/)  +  4(2/2  -  2)  -  3y  +  2  =  0; 
whence,  j/^  +  4  2/*  —  6  2/  —  6  =  0. 


422  ADVANCED  ALGEBRA. 

EXERCISE     107. 

Solve  : 

Z.  QC^  —  dx^-\-^x  —  l  =  0  4.  ic*  +  3a;=^  —  3a;— 1  =  0 

5.  2a;*-5a;3  +  4a;2~5a;  +  2  =  0 

6.  a;s  +  5a;*  +  10a;3  +  10:z;2_j_5^_^l_0 

7.  6a;5-ir*-432;-''  +  43a;2  +  ir-6  =  0 

8.  bx'>-\-ll7^-^%x?-^^x^-^llx-{-b  =  0 


Reduction  of  Binomial  Equations. 

785.  A  Binomial  Equation  is  an  equation  of  two  terms, 
one  of  which  is  absolute  ;  as,  a;"  ±  «  =  0. 

786.  Every  Mnomial  equation  can  te  reduced  to  the 
form  2/**  ±  1  =  0. 

Demonstration :  Take  the  general  binomial  equation  a;*  ±  a  =  0. 


Put  ^"^  for  X, 

y 

a 

r 

+  a  =  0 ;  whence,  2/"  ±  1  =  0. 

787.  r  ±  1 

=  0 

is  a  recurring  equation,  and  may  be  so 

solved. 

Illustrative  Solutions.— 1.  Solve  a:*  + 1  =  0.             (A) 

Divide  by  x^, 

^^  +  ^  =  0                                         (1) 

Put 

^^\  =  y                            (3) 

Square 

x^  +  2+l,  =  y^ 

Transpose, 

^'  +  ^,  =  y'-^ 

Substitute  in 

(1), 

y^-2  =  0 

Factor,    (y  +  ^2 ),  (y  -  a/s")  =  0 ; 
whence,  y  —  ±  \/2. 

Substitute  in  (2),  x  +  -=  ±  \/2  (3) 

X 

whence,    x  =  ^ {\/2  ±  ^/'^),  or  -  ^ {^ T  ^-2)- 


chapter  xiii. 
determij^a:n'ts  akb  probabilities. 


Introduction. 
788.  In  the  polynomial 

«!  1)2  Ci  —  «i  ^3  C2  +  «8  ^3  ^1  —  «2  ^1  ^3  +  ^3  ^1  ^2  —  ^3  ^2  ^1  ^        (-^) 

it  will  be  seen  : 

1.  That  the  letters  a,  h,  and  c  of  each  term  are  ar- 
ranged in  natural  order. 

2.  That  the  subscript  figures,  1,  2,  and  3,  are  dis- 
tributed among  the  letters  in  the  six  different  terms  in 
as  many  ways  as  possible,  using  all  in  each  term  and 
making  no  repetitions. 

3.  That  the  first  term  contains  no  inversions  of  sub- 
script figures,  they  advancing  in  natural  order  from  left  to 
right ;  the  second  term  contains  one  inversion,  3  standing 
before  2 ;  the  third  term  contains  two  inversions,  2  and  3 
both  standing  before  1 ;  the  fourth  term  contains  one  in- 
version, 2  standing  before  1 ;  the  fifth  term  contains  two 
inversions,  3  standing  before  1  and  2  ;  and  the  sixth  term 
contains  three  inversions,  3  and  2  both  standing  before  1, 
and  3  standing  before  2. 

4.  That  in  the  positive  terms  there  is  an  even  number 
of  inversions  (zero  being  regarded  an  even  number),  and 
in  the  negative  terms  there  is  an  odd  number  of  inver- 
sions. 


424 


ADVANCED  ALGEBRA. 


(B) 


789.  If  we  now  arrange  the  nine  different  quantities 
found  in  (A)  in  a  square,  as  follows  : 

«i    a2    a^ 
h     h     h      ; 

Ci      Cg      C3 

form  all  the  possible  products  of  them  taken  three  to- 
gether, using  in  each  product  one  and  only  one  from  each 
row,  and  one  and  only  one  from  each  column ;  arrange 
the  factors  of  the  products  in  the  natural  literal  order ; 
consider  those  products  positive  which  have  an  even  num- 
ber of  inversions  of  subscript  figures,  and  those  negative 
which  have  an  odd  number ;  and  take  the  algebraic  sum 
of  these  products,  we  will  have  : 

«i  ^2  C3  —  a^  J3  (?2  +  «3  ^3  Ci  —  «3  li  C3  +  a^  bi  <?2  —  %  ^2  Ci .  (A) 
Therefore,  form  (B)  may  be  taken  as  the  representative 
of  form  (A),  and  when  so  taken  it  is  called  a  determinant, 
and  (A)  its  development. 

790.  Definition. — A  Determinant  is  any  n^  quantities 
arranged  in  a  square,  as  follows  : 


at 

«2      «3 

h 

\       h 

Cx 

C%       C3 

(C) 


Pi      P2      PZ Pn 

and  interpreted  to  denote  the  algebraic  sum  of  all  the 
products  that  may  be  formed  by  taking  one  and  only  one 
quantity  from  each  row,  and  one  and  only  one  from  each 
column  ;  arranging  the  letters  of  each  product  in  the  natu- 
ral literal  order,  and  regarding  all  products  positive  that 
have  an  even  number  of  inversions  of  subscript  figures, 
and  all  negative  that  have  an  odd  number  of  such  inver- 
sions. 


DETERMINANTS.  425 

791.  The  quantities  contained  in  a  determinant  are 
called  the  elements  of  th^  determinant. 

792.  Determinants    are    divided    into    orders,   named 

second,  third, nila.,  accordingly  as  they  contain  2^,  3^, 

n^  elements. 


Thus, 


is   a   determinant  of    the   second 


h    h 

order.     Form  (B)  is  a  determinant  of  the  tMrd  order,  and 
form  (C)  a  determinant  of  the  ntli  order. 

793.  The  diagonal  joining  the  upper  left-hand  element 
with  the  lower  right-hand  element  is  called  the  'principal 
diagonal ;  and  the  one  joining  the  upper  right-hand  ele- 
ment with  the  lower  left-hand  element  the  secondary 
diagonal. 

794.  The  product  of  all  the  elements  along  the  principal 
diagonal  is  called  the  principal  term  of  the  development. 

795.  If  the  elements  on  the  principal  diagonal  are 
known  in  order,  the  entire  determinant  may  be  written ; 
hence  it  is  that  a  determinant  is  often  expressed  by  a 
modified  form  of  the  principal  term  of  its  development ; 
as,  \_a^lzc^ ^J,  or  S(±ai^2^3 i?«)- 

796.  It  is  evident  that  there  are  as  many  terms  in  the 
development  of  a  determinant  of  the  wth  order  as  there 
are  permutations  of  n  things  taken  all  together,  or  \n. 


Properties  of  Determinants. 

797.  If  we  rearrange  the  factors  of  the  terms  in  form 
(A)  so  as  to  place  the  subscripts  in  natural  order,  we  shall 
have 

«!  ^8  ^3  —  «i  Cg  ^3  +  Ci  ttg  ^3  —  5i  ^2  <^3  +  ^1  ^2  <^3  "  ^1  ^3  «3'       (^i) 


4:26  ADVANCED  ALGEBRA. 

It  will  be  observed  that  in  form  (Aj), 

1.  The  value  of  each  term  and  of  the  entire  polyno- 
mial is  the  same  as  in  form  (A). 

2.  The  first  term  contains  no  literal  inversion ;  the 
second  term  contains  one,  c  standing  before  h ;  the  third 
term  contains  two,  c  standing  before  both  a  and  b ;  the 
fourth  term  contains  one,  h  standing  before  a;  the  fifth 
term  contains  two,  h  and  c  both  standing  before  a ;  and 
the  sixth  term  contains  three,  c  and  h  both  standing  before 
a,  and  c  before  h 

3.  The  terms  which  contain  an  even  number  of  literal 
inversions  are  positive,  and  those  which  contain  an  odd 
number  negative, 

798.  If  we  now  interchange  the  rows  and  columns  in 
(B),  giving  us  the  form 

ttx    hi    Ci 

«2    h    ^2      ;  (D) 

%    h    ^3 

make  all  the  possible  products  of  three  elements,  using, 
each  time,  one  and  only  one  from  each  row,  and  one  and 
only  one  from  each  column ;  arrange  the  factors  of  the 
products  so  that  the  subscripts  stand  in  natural  order ; 
consider  those  products  positive  which  have  an  even  num- 
ber of  literal  inversions,  and  those  negative  which  have  an 
odd  number ;  and  take  the  algebraic  sum  of  these  prod- 
ucts, we  shall  have 

«1  ^2  ^3  "~  ^1  ^2  ^3  +  ^1  ^2  ^3  —  ^1  ^2  ^3  +  ^1  <^2  ^3  "  ^1  ^2  ^3  •      (-^-i) 

This  shows  that  in  a  determinate  of  the  third  order  an  in- 
terchange of  rows  and  columns  does  not  change  the  value. 

Is  this  law  true  for  a  determinant  of  the  nth  order  ? 

1.  It  is  evident  that  the  number  of  terms  in  the  de- 
velopment of  both  forms  is  the  same,  each  being  [w. 


PROPERTIES  OF  DETERMINANTS.  427 

2.  Each  term  in  the  development  of  either  form  has  a 
corresponding  term  of  equal  numerical  yalue  in  the  devel- 
opment of  the  other  form,  because  both  developments  con- 
tain all  the  possible  products  of  n  elements  that  can  be 
formed  from  the  n^  elements  by  taking  one  and  only  one 
from  each  row  and  one  and  only  one  from  each  column. 

3.  The  signs  of  the  corresponding  terms  toill  be  the 
same.  For  the  number  of  literal  inversions  in  a  term  of 
the  second  development  is  equal  to  the  number  of  sub- 
script inversions  in  the  corresponding  term  of  the  first 
development,  as  will  readily  appear  from  the  fact  that,  if 
a  subscript  in  any  term  of  the  first  development  follows  r 
subscripts  greater  than  itself,  then,  in  the  second  develop- 
ment, the  letter  containing  this  subscript  must  precede  r 
letters  antecedent  to  it  in  the  natural  order.     Therefore, 

JPrin.  1. — An  interchange  of  rows  and  columns  in  a 
determinant  of  any  order  does  not  change  the  value  of  the 
determinant.  

799.  In  form  (A)  and  in  form  (Ai)  the  second  term 
equals  minus  the  first  term  with  the  subscripts  of  h  and  c 
interchanged ;  the  third  term  equals  minus  the  second 
term  with  the  subscripts  of  a  and  c  interchanged ;  and  so 
on,  showing  that  any  term  in  the  development  of  a  deter- 
minant of  the  third  order  equals  minus  some  other  term  in 
the  development  with  the  subscripts  of  two  factors  inter- 
changed. 

Is  this  law  true  for  the  development  of  a  determinant 
of  the  nth  order  ? 

1.  It  is  evident  that,  if  Pcr^^  be  any  term  in  the  de- 
velopment of  a  determinant  of  the  nth  order,  then  will 
Fcrnhhe  numerically  another  term  of  the  development; 
because  P  in  both  instances  is  the  product  oi  n  —  2  ele- 


4:28  ADVANCED  ALGEBRA. 

ments,  none  of  which  are  taken  from  rows  c  and  Jc,  and 
none  from  columns  r  and  w;  and  c^h^  and  c^h,  are 
different  elements  taken  from  these  rows  and  columns 
and  combined  with  P.  Therefore,  the  products  are  not 
identical. 

2.  The  signs  of  the  original  and  the  derived  terms  are 
always  opposite.     For, 

(1)  Suppose  the  two  subscripts  interchanged  to  be  con- 
secutive. Let  the  original  term  be  P  c^  h^  Q,  and  the  de- 
rived term  P  c„  k^  Q.  Since  m  and  n  follow  all  the  sub- 
scripts contained  in  P  and  precede  all  contained  in  Q,  an 
interchange  of  them  can  not  affect  the  number  of  inver- 
sions they  make  with  the  subscripts  of  either  P  or  Q ;  but 
such  an  interchange  will  either  change  a  natural  into  an 
inversion  or  an  inversion  into  a  natural,  either  of  which 
will  evidently  cause  a  change  of  sign, 

(2)  Suppose  the  two  subscripts  interchanged  to  be  non- 
consecutive.  Let  the  original  term  be  Pc^Q  Jc^  R,  and 
the  derived  term  P c^Qhn R.  Suppose  Q  to  contain  q 
subscripts.  Let  m,  in  the  original  term,  interchange  con- 
secutively with  each  of  the  subscripts  in  Q  and  with  the 
subscript  of  c,  then  will  it  make  q-{-l  interchanges  before 
it  becomes  the  subscript  of  c,  n  will  now  be  the  subscript 
of  the  first  element  in  Q.  Let  it  now  interchange  con- 
secutively with  each  of  the  remaining  subscripts  in  Q  and 
with  the  subscript  of  h ;  then  will  it  make  q  interchanges 
before  it  becomes  the  subscript  of  Ic.  Therefore,  for  the 
two  subscripts  of  c  and  Jc  in  the  original  term  to  inter- 
change there  must  be  made  2q-{-l,  or  an  odd  number  of 
consecutive  interchanges,  each  one  of  which  will  cause  a 
change  of  sign  (1)  in  the  entire  term.  Therefore,  the  sign 
of  the  term  will  be  changed.     Therefore, 


PROPERTIES  OF  DETERMINANTS,  429 

Prin.  2. — If  tioo  subscripts  he  interchanged  in  any 
term  of  the  development  of  a  deter miiianty  another  term 
of  the  development  will  be  obtained  whose  sign  is  opposite 
to  that  of  the  original  term. 


800.  If  we  let  Ph,^QKR  be  a  term  in  the  develop- 
ment of  a  determinant,  then  will  Ph^QKR  be  the  term 
formed  by  the  elements  which  occupy  the  same  places,  if 
rows  h  and  h  be  interchanged,  and  will  have  the  same 
sign.  But  Ph„,QKR  is  also  a  term  of  the  development 
of  the  original  determinant,  and  has  there  an  opposite  sign 
to  Ph„,Q KB  [P.  2].     Therefore, 

Prin,  3, — Interchanging  two  rows  in  a  determinant 
changes  the  sign  of  the  determinant. 

Cor.  1, — Interchanging  two  columns  in  a  determinant 
changes  the  sign  of  the  determinant. 


801.  It  is  evident  that  if  two  columns  or  two  rows  of  a 
determinant  are  in  every  respect  alike,  an  interchange  of 
them  would  not  affect  either  the  form  or  value  of  the  de- 
terminant. But,  according  to  Principle  3,  the  sign  of  the 
value  would  be  changed.  Now,  both  these  statements  can 
be  true  only  when  the  value  of  the  determinant  is  zero. 
Therefore, 

Prin,  4, — A  determinant  that  has  two  rows  or  ttvo  col- 
umns identical  equals  zero. 


802.  Since  every  term  in  the  development  of  a  deter- 
minant contains  one  factor  and  only  one  from  each  row 
and  one  and  only  one  from  each  column,  it  follows  that, 

Prin,  5. — Multiplying  or  dividing  all  the  elements  of 
one  row  or  one  column  of  a  determinant  by  any  quantity 
multiplies  or  divides  the  determinant  by  that  quantity. 


430 


ADVANCED  ALGEBRA. 


Cor,  1, — Changing  the  signs  of  all  the  elements  in  any 
row  or  column  changes  the  sign  of  the  determinant. 

Car,  2, — If  two  rows  or  tiuo  columns  of  a  determinant 
differ  only  hy  a  common  factor,  the  value  of  the  deter- 
minant is  zero, 

803.  Definitions. — If  any  number  of  rows  and  the  same 
number  of  columns  be  deleted  (stricken  out)  of  a  deter- 
minant, the  remaining  elements,  taken  in  order,  form  a 
determinant  called  a  minor,  and  the  elements  common  to 
the  deleted  rows  and  columns  form  another  minor.  These 
minors  are  said  to  be  complementary. 

Thus,  in  the  following  determinant  of  the  fourth  order. 


h 


-do 


^T 


d, 
-dr 


the  complementary  minors  are 
h    di 
h    ds 


and 


«2 
t?4 


804.  If  a  single  row  and  a  single  column  be  deleted, 
the  remaining  minor  is  called  the  principal  minor,  and  it, 
together  with  its  complementary  minor,  which  in  this  in- 
stance is  a  single  element,  are  called  cofactors. 


805.        Problem.    To  develop  a  determinant. 

Let  it  be  required  to  develop 

«i    «8    ^3    a^ 
h    h    h    h 

C\        Cg        ^3        C4 

d\    d^    dz    di^ 
into  a  series  of  determinants  of  a  lower  order. 

Let  ^1,  Ai,  At^  and  A^  represent  respectively  the  cofactors  of 
«i ,  «a ,  as ,  and  a^ .    Then,  it  is  readily  seen  that  ail  the  terms  in  the 


PROPERTIES  OF  DETERMINANTS. 


431 


development  containing  the  factor  a^  are  formed  from  ai  and  its  co- 
factor  Ai ,  and  the  sum  of  these  terms  is  ai  Ai .  Similarly,  the  sum 
of  all  the  terms  containing  a^  is  a^  At,  the  sum  of  all  the  terms  con- 
taining aa  is  tta  Aa,  and  the  sum  of  all  the  terms  containing  a*  is 
at  At .  Now,  in  each  term  of  a^  A^  there  occurs  one  more  inversion 
than  in  each  term  of  ai  Ai ,  since  a  subscript  2  will  precede  a  subscript 
1 ;  similarly,  in  each  term  of  as  -4s  there  occur  two  more  inversions 
of  subscripts  than  in  ai  Ai,  and  in  each  term  of  a*  A^  there  occur 
three  more  inversions  of  subscripts  than  in  Ui  Ai. 
Therefore, 


«!    tti   az    at 
b\    bi    bz    bi 

C\      Ci      Cz      C4 

di    di   dz   d^ 

=  ai 

bi    bz 
Ci    Cz 
di   dz 

&4 
C4 

—  «2 

b\    bz 
Ci     Cz    •« 

di    dz 

74          + 
^4 

bi    bi    &4 

bi    bi    bz 

Oz 

c 
d 

1      Ci     C4 
I     di    d. 

—  ai 

ci    Ci    Cz 
di    di  dz 

Therefore, 

Mtde, — Multiply  each  element  of  the  first  row  hy  its 
cof actor,  making  the  products  alternately  plus  and  minus, 
and  talce  the  algebraic  sum  of  the  results, 

806.  Scholium, — The  successive  application  of  this  rule 
will  eventually  make  the  full  development  of  any  deter- 
minant depend  upon  the  development  of  a  determinant  of 
the  second  order.     Thus, 

h  h   h 


Cz     C^     Ci 

dz  ds  t?4 


=  h 


C3   64 
ds  di 


-K 


Cz     Ci 

dz  di 


+  *4 


Cz     C3 

do  d% 


h  (^3  di  —  d  ds)  —  bs  (cz  di  —  C4  dz)  +  h  (Cz  d^  —  c^  dz) 


Ulnstrative  Example.— Find  the  value  of 


Solution : 

2  3    4 

3  2    4 

4  3    2 

2(2x2 


-3 


+  4 


4  X  3)  -  3  (3  X  2 


4x4)  +  4(3  x3-2  x4)  = 

-  16  +  30  +  4  =  18. 


432 


ADVANCED  ALGEBRA. 


EXERCISE     lOa 


Find  the  value  of  : 


10. 


13. 


3  15 

2. 

4  3  2 

3. 

4  2  7 

5  14 

16  4 

2  4  5 

a  0  5 

5. 

m  0  n 

6. 

-a  d  0 

m  p  0 

a  b  ( 

0  p  n 

2  3  2  3 

8. 

3  2  3  2 

2  3  3  2 

3  2  2  3 

2  3  12 

3  2  14 
2  3  12 
14  3  2 


11. 


3  4  12 

14. 

6  8  2  4 

12  3  4 

2  4  6  8 

1 

2  3  4 

9. 

2 

-13  4 

3 

2  13 

1 

2  10 

2  3  4  2 

12. 

2  3  4  2 

2  3  4  2 

2  3  4  2 

3-1      4 
2       1-3 

4      2      7 

h-\-c  c  h 
c  a-\-c  a 
h      a    a-\-h 

0  2  3  4 

2  0  3  4 

3  2  0  2 

4  12  0 

a  h  c  d 

&  f  9  ^ 
a  I  c  d 
i  k  I  m 


12  0  1 

15. 

2  4  0  1 

3  6  0  1 

4  8  0  1 

a 

h 

c 

d 

a 

b 

c 

d 

n 

n 

n 

n 

a 

c 

h 

d 

c 

a 

d 

I 

Additional  Properties  of  Determinants. 

807.  JPHn,  6, — If  every  element  of  one  row,  or  column, 
of  a  determinant  is  a  binomial,  the  determinant  can  be 
expressed  as  the  sum  of  two  other  determinants,  one  of 
which  is  derived  from  the  original  determinant  by  drop- 
ping the  second  terms  of  the  binomials  and  the  other  by 
dropping  the  first  terms. 

Demonstration. — Each  term  of  the  development  contains  one  and 
only  one  of  the  binomial  elements  as  a  factor.  Therefore,  each  term 
of  the  development  can  be  separated  into  two  terms,  one  of  which  is 
the  first  term  of  the  binomial  factor  times  the  remaining  factors  of 


PROPERTIES  OF  DETERMINANTS. 


433 


a 

b  +  c 

b 

a 

b 

b 

a 

c 

b 

b 

a  +  c 

a 

= 

b 

a 

a 

+ 

b 

c 

a 

c 

a  +  b 

b 

c 

a 

h 

c 

b 

b 

the  term,  and  the  other  the  second  term  of  the  binomial  factor  times 
the  remaining  factors  of  the  term.  The  sum  of  the  component  parts 
that  contain  the  first  terms  of  the  binomial  elements  will  form  a  de- 
terminant which  is  independent  of  the  second  terms  of  the  binomial 
elements,  and  the  sum  of  the  component  parts  that  contain  the  second 
terms  of  the  binomial  elements  will  form  a  second  determinant  which 
is  independent  of  the  first  terms  of  the  binomial  elements. 


Thus, 


808.  Cor.  1, — If  every  element  in  any  row,  or  column, 
consists  of  m  terms,  the  determinant  can  be  expressed  as 
the  sum  of  m  qther  determinants. 

809.  Oor.  2. — If  the  elements  of  r  rows,  or  columns, 
consist  of  a,h,Cf m  terms  respectively,  the  determi- 
nant can  le  expressed  as  the  sum  of  abc m  determi- 
nants. 

810.  Scholium,, — These  truths  are  of  value  in  reducing 
a  determinant  when  one  or  more  of  the  derived  determi- 
nants reduce  to  zero.     Thus, 


a     h^ 

•a-\-c    c 

abc 

a  a 

c 

ace 

d    e+d^f  f 

= 

d  e  f 

+ 

d  df 

+ 

df  f 

g    h-\-g-\-h    h 

g  h  k 

9  9^ 

g  k  k 

a  h  c 

ale 

d  e  f 

+  0  +  0  [801,  P.]  = 

d  e  f 

g  h  h 

9 

h  I 

; 

811.  Prin,  7* — If  to  all  the  elements  of  any  row,  or 
column,  he  added  equimultiples  of  the  corresponding  ele- 
ments of  any  other  row,  or  column,  the  value  of  the  deter- 
minant will  remain  unchanged. 

Demonstration. — Consider  a  determinant  of  the  third  order.    Thus, 


Prove 


«1 

«2 

^3 

h 

h 

h 

Ci 

Cz 

Cz 

434 


ADVANCED  ALGEBRA. 


Demonstration. 


a\  +  paz    aa    (H 

ai    ai 

3^8 

paz 

as    oz  1 

h    +i?&3       &2      ^3 

= 

b\     &2    bz 

+ 

p  bz    bi    bz 

Ci   +pcz      Ci     Cs 

C\      Ci      Cz 

pcz      Ci      Cz 

a\    a<i,    as 

dX      di      Clz 

bx     h    h 

+  0  [802,  Cor.  2]  = 

b\     bi     bz 

C\       Cn      Cs 

C\        C2 

Cz 

The  method  of  proof  employed  in  this  example  is  general,  and  is, 
therefore,  applicable  to  a  determinant  of  any  order. 

812.  Car, — It  may  also  be  shown  that 

ai-{-pa2-\-qas    a^ 

Ci  -\rpc2-\-q  Cg     -cg 


«3 


Oz 


«3 
^3 


;  etc. 


«i    a^ 

Ci       C2 

That  is. 

To  all  the  elements  of  any  row,  or  column,  may  be  added 
equimultiples  of  the  corresponding  terms  of  a  second  row, 
or  column,  and  again  equimultiples  of  the  corresponding 
terms  of  a  third  row,  or  column,  etc. 

813.  Scholium, — This  principle  is  of  practical  value 
in  the  reduction  of  a  determinant,  if,  by  its  application, 
two  rows,  or  columns,  can  be  made  identical,  or  one  of 
them  a  multiple  of  the  other.     Thus, 

5  +   8  + 11     8  11  24    8  11 

6+9  +  12     9  12     =     27     9  12     =0 
7  +  10  +  13  10  13  30  10  13 

[802,  Oor.  2].     It  may  also  be  used  to  simplify  a  complex 
determinant. 


5    8  11 

6     9  12 

= 

7  10  13 

814.  Prin,  8. — If  the  elements  of  one  column  of  a  de- 
terminant be  multiplied  by  the  cof actors  of  {he  correspond- 
ing elements  of  another  column,  the  sum  of  the  products, 
taTcen  alternately  plus  and  minus,  is  zero. 

Demonstration. — Take  two  determinants,  alike  in  every  respect, 
except  that,  in  the  second,  the  gth  column  is  identical  with  the  ^th 
column.    Now,  in  the  first,  the  sum  of  the  products  of  the  elements 


PROPERTIES  OF  DETERMINANTS. 


435 


in  the  pih.  column  and  the  cofactors  of  the  corresponding  elements  of 
the  ^th  column,  when  the  products  are  taken  alternately  plus  and 
minus,  is  apAq  —  bpBg  + 

The  value  of  the  second  determinant  is 

agAg-bgBq  +  ..,.  [805,  R.]  =  0  [801,  P.J. 

But  it  is  evident  that  aq ,  bq , are  identical  with  Up,  bp,  .... 


Therefore, 


apAq  —  bpBq  + =0. 


EXERCISE     109. 


Find  the  value  of 


3  2 
5  3 

7  4 

0 

8 

6 

-8 


3 
4 
5 

-1 
5 
9 
4 


6. 

a+p 
b-hp 
o+p 

7.  Prove 

8. 

Prove 

-2 
2 
8 


-4 

6 

-9 


2       0 

2  2 

3  -3 


b  +  q 
c-^q 
a-\-q 

h-\-c 
q-\-r 

aJ^  —  be 


1 
1 
1 
1 

c  -\-r 
a-\-r 
b-^-r 

c-\-a  a-\-b 
r^-p  p-\-q 
z-^x    x-{-y 

(?  —  ab 


+ 


y-z 
z  —  y 

a  —p 
b  —  p 
c  —p 


=  2 


3. 

12  -5 
-3      4 

7      0 

X  — 

z  — 

-z    x-y 

•  X    y  —  X 

X    x-\-y 

b-q 
c-q 
a  —  q 

a  b 
p  q 
X    y 


(^  —  ab    b^  —  ac 


ciSabc-a^-b^-c") 


Multiplication  of  Determinants. 


B15.  Lemma,— 

ai     bi    Ci     I 

m 

n 

«2    ^2    02    p 
%     bs     c-3     s 
0      0     0     a^i 
0      0     0     a^g 

t 

yi 
y2 

r 
u 

^2 

= 

«2    ^2    ^2 
«3    ^3    ^3 

0      0     0     a;3 

^3 

2^3 

X 


Xi  yi  Zi 

^2  y2  % 
^^  yz  z^ 


436 


ADVANCED  ALGEBRA, 


Demonstration.— 

ax  h\  ci  I  m  n 

ai  h  Ci  p  q  r 

as  bs  Cz  s  t  u 

0  0  0  x\  y\  Zi 

0  0  0  a:2  2/2  ^2 

0  ()  0  xz  yz  Zz 


+  az 


=  ai 


bi  Ci  p    q    r 

5i 

c\  I     m  n 

bz   Cz   8     t      u 

&3 

Cz   s     t     u 

0    0   a;i  yi  zi 

—  as 

0 

0    xi  yx  zx 

0    0   Xi  yi  Zi 

0 

0     Xi    2/2    ^2 

0    0   xz  yz  zz 

0 

0    Xz  yz  Zz 

bx  cx    I    m  n 

bi  Ci  p    q  ■  r 

0    0    a;i  2/1  Zx 

=  ai  &2 

0       0     iCs    2/2    ^2 

0    ^    Xz  yz  Zz 

Cz  s     t     u 

0  a;i  2/1  ^1 

0     Xi    2/2    2^2 

—  ai&8 

0  Xz  yz  Zz 

Ci 

p    q    r 

0 

xx  yx  Zx 

0 

Xi    2/2    Zi 

0 

Xz  yz  Zz 

—  tti  bx 


—  az  bi 


Cz  s     t     u 

0   Xx  yx  Zx 
0    Xi  yi  Zi 

+  a2&3 

0   Xz  yz  Zz 

Cx  I    m  n 

0   ici  2/1  2;i 
0    Xi  yi  Zi 

+  aa&i 

0    Xz  yz  Zz 

Ci 

p    q    r 

0 

Xx  yx  Zx 

0 

Xi  yi  Zi 

0 

Xz  yz  Zz 

Cl 

I    m  n 

0 

Xx  yx  Zx 

0 

Xi  yi  Zi 

0 

Xz  yz  Zz 

Xx  yx  Zx 

ax  bi  Cz 

Xi  yi  Zi 

Xz  yz  Zz 

—  ax  bz  Ci 


Xx 

2/1 

Zx 

Xi 

2/2 

Zi 

Xz 

2/3 

Zz 

Xx  yx  Zx 

Xi 

2/1  Zx 

Xx  yx  Zx 

ai  bx  Cz    Xi  yi  Zi 

+  ai  bz  Cx 

Xi  yi  Zi 

+  az  bx  Ci 

Xi  yi  Zi 

Xz  yz  Zz 

Xz  yz  Zz 

Xz  yz  Zz 

Xx  yx  Zx 

ax  bi  C\ 

Xx  yx  Zx 

az  bi  Cx 

Xi  yi  Zi 

= 

ai  bi  Ci 

X 

Xi  yi  Zi 

[789,  (A)]. 

Xz  yz  Zz 

az  bz 

Cz 

Xz  y»  Zz 

Scholinm. — It  will  be  seen  that  the  elements  ?,  m,  n,  p,  q,  r,  s,  t,  u 
of  the  first  member  do  not  appear  in  the  second  member.  This  is  due 
to  the  fact  that  in  the  development  of  the  first  member  all  the  terms 
containing  these  elements  eventually  vanish. 

816.  Problem.  To  find  the  product  of  two  determinants 
of  any  order  in  terms  of  a  determinant  of  the 
same  order. 

Example. — Eind  the  product  of  : 

«1      ^1      Ct  Xi      j/i      Zx, 

«2       hz      Cz  X  Xz      yz      ^2 

«3       ^3       ^3  ^3      Vz      % 


MULTIPLICATION  OF  DETERMINANTS.        437 


Explanation : 


a\  hi  C\ 

xi  yi  zx 

tti  bi  Ci 

X 

Xi  2/2  22 

az  h  Cz 

Xi  ys  zz 

a\  hx  C\   —\       0 


0 

0-1  0 
0       0   -1 

Xx  yx  Zx 
Xi  2/2  Zi 
xz     yz     zz 

-1 
0- 


[815] 


0  0  0-100 

0  0  0  0-10 

0  0  0  0     0-1 

aiXi  +  a%yi  +  aaZi  biXi  +  b^yi  +  baZi  CiXi  +  dyi  +  dZi  Xi  yt  Zi 
aiXi-\:aiyi  +  aaZi  biXt  +  btyi  +  baZi  CiXt  +  c^yi  +  CaZa  Xi  yi  z^ 
axXz-\-aiy%-^aaZa  biXa  +  b^ya  +  bsZa  CiXa  +  C^ya  +  CaZa     Xa    ya     Za 

(This  last  form  is  derived  from  the  preceding  by  adding  to  the 
first  column  ax  times  the  4th  column  +  a^  times  the  5th  column  + 
az  times  the  6th  column ;  to  the  second  column,  bx  times  the  4th  + 
&2  times  the  5th  ^  bz  times  the  6th ;  and  to  the  third  column,  Cx  times 
the  4th  +  c%  times  the  5th  +  Cz  times  the  6th  [812]) 

ax  Xx  +  as  2/1  +  (H  Zx    Ix  Xx  +  &2  yi  +  bz  zx    Cx  Xx  +  62^1  +  Cz  Z\ 
ax  Xi  +  a^yi  +  az  z%    bx  Xi  +  b^y^  +  bz  z^    Cx  Xi  +  Co  2/2  +  Cz  22 
ax  xz  +  a^yz  +  az  Zz    bx  xz  +  b^yz  +  bz  Zz    C\  X3  +  c^yz  +  cz  Zz 
[805,  R.]. 

Let  the  student  observe  how  the  elements  of  the  first  column  of 
the  product  are  derived  from  the  elements  of  the  multiplicand  and 
multiplied ;  then,  how  the  elements  of  the  second  and  third  columns 
are  found.  The  laws  which  he  will  observe  are  general  for  determi- 
nants of  any  order. 

Example. — Sliow  that,  according  to  the  laws  above  ob- 
served. 


X 


^1  yi 
^2  Vz 


ax  Xz  +  ttz  1/2    h  ^%  +  ^2  ^2 


EXERCISE     110. 


Find  the  value  of  : 


2  3 

1  2 


X 


3  1 
2  4 


3. 

12  3 

3  0  1 

2  3  1 

X 

2  13 

13  2 

12  3 

1  3 

2  -2 


X 


3  0 

1  2 


2  0  3 

10  0 

3  10 

X 

2  2  2 

2  0  3 

13  2 

438  ADVANCED  ALGEBRA. 

Applications. 

817.  I.  Solution  of  Simultaneous  Equations  of  the 
First  Degree, 

1.  Solve  aiX-\-'biy  =  ri  (A) 

«2^+^2  2/  =  ^2  (B) 

Solution :  Multiply  (A)  by  ^i  and  (B)  by  —A^  ,  in  which  ^i  and 
Ai  are  the  eofactors  of  ai  and  as  in  the  determinant  [  ai  h  ],  and 
take  the  sum, 

{ai  Ai  —  tti  Aii)x  +  {hi  Ai  —  h  Ai)y  =  ri  Ai  —  r2  A^  (C) 

Now,    hi  Ai-hA.i  =  0  [814,  P.].    Therefore, 
(ai  A\  —  tti  Ai)  X  =  ri  Ai  —  n  An ;  whence, 

aiAi  —  a^Ai      [  ai  os  J  ^      '       -■ 
Again,  multiply  (A)  by  Bi  and  (B)  by  — -Bs ,  in  which  Bi  and 
Bi  are  the  eofactors  of  h\  and  B^ ,  and  take  the  sum, 

(ai  Bi  -  tti  Bci)x  +  (bi  Bi  -h  Bi)y  =  n  Bi  -  n  Bi  (D) 

Now,    ai  Bi-a2Bi  =  0  [814,  P.].    Therefore, 


2.  Solve  aiX-\-biy-\-CiZ  =  ri  (A) 

«2^+^2«/+^2^=^2  (B) 

a3^  +  hy  +  CsZ  =  rs  (C) 

Solution :  Multiply  (A)  by  J.i ,  (B)  by  —Ai,  (C)  by  J3 ,  in  which 
Ai ,  Ai ,  and  ^Is  are  respectively  the  eofactors  of  ai ,  as ,  and  as  in 
the  determinant  [  ai  fts  Cs  ],  and  add  the  resulting  equations, 

(ai  Ai  —  tti  Ai  +  as  A3  )x  +  (bi  Ai  —  bi  At  +  bzAz)!/ 

+  (ci  Ai  —  Ci  Ai  +  C3  Az)z  —  n  Ax  —  Ti  Ai  +  r%  Az 

Now,  the  coefficients  of  y  and  z  vanish  [814,  P.]'; 

^  ^  nA,-r,A,^nA,  ^  [n  t,  ..] 

ai  J.1  —  as  J-s  +  as  -43       [  ai  02  C3  J 
Then,  by  symmetry, 

^       [  ai  62  C3  ]  [  ai  62  cz  ] 

In  a  similar  manner  it  may  be  shown  that,  if  we  take  n  equations 
of  the  first  degree  of  the  form  of  ai  ic  +  61 2/  +  Ci  2  + . . . .  =  ri ,  mul- 


MULTIPLICATION  OF  DETERMINANTS.        439 

tiply  the  first  by  A\ ,  the  second  by  —A2 ,  the  third  by  Az , ,  in 

which  Ai ,  A2,  A3, ,  are  the  cofactors  ot  ai ,  a^ ,  as , ,  and 

take  the  sum,  the  coefficients  of  all  the  unknown  quantities,  except  x, 
will  vanish,  and  we  shall  have 

X  =  F — ^-^ — ^  j  ;  and,  by  symmetry, 

[ai  h  cz ]'  •" 

[a,   r,  ^3....]   ^^^^    Therefore, 
^       [ai   bi  cz ]' 

Principle, — Any  unknown  quantity  in  a  complete  sys- 
tem of  simultaneous  equations  of  the  first  degree  equals  a 
fraction  whose  denominator  is  a  determinant  formed  from 
the  coefficients  of  the  terms  of  the  equations  taken  in  order, 
and  whose  numerator  is  formed  from  the  denominator  hy 
replacing  the  coefficients  of  the  unknown  quantity  hy  the 
corresponding  right  members  of  the  equations, 

EXERCISE     111. 

Solve  : 

I.  3a;-|-2y  =  16  2.6x  —  ^y=    6 

'Zx  —  ^yz^   2  2x-\-5y  =  21 

3.   ax-]-by  =  c  4.  (a-\-b)x  —  (c-\-d)y=:  m 

mx-\-ny  —  d  (a-^b)x-{-  (c  —  d)y  =  n 

5.  2x-\-dy  —  2z=5  6.     x--y-\-z=6 

dx-2y-{-4:Z=16  Sx-{-6y  —  Sz=14: 

4:X  —  3y—    z=—6  2x-{-4:y  +  Sz  =  20 

1.  ax-\-by-\-  cz  =  d  s,  {a-\-b)x-\-by-\-az  =  m 

cx-\-by-\-az  =  e  ax-\-{a-{-b)y-{-bz  =  n 

bx-\-cy-\-az  =  h  bx-\-ay-\-{a-\-b)z  =  r 

9.  x-\-y-{-z-\-u  =  14:  10.  ax-{-by+cz=m 

X  —  y~^z  —  u=  —  2  bx-^  cy-\-au  =  n 

x-\-y  —  z  —  u=  —  4:  cx-\-az-\-bu=p 

X  —  y  —  z-\-u=    0  ay-{-bz-\-cu  =  q 

11,  2x^Sy  +  2z-{-    u  =  -12 

3x-[-2y-dz-{-2u  =  12  - 

x  —  Sy-\-4:Z  +  3u=  —  24: 
2x-]-2y-dz-4cU=z2S 


440  ADVANCED  ALGEBRA. 

12.  ax-\-'by-{-cz-\-du=p 
ax  —  dy-\-cz  —  du  =  q 
ax-\-'by  ^  cz  —  du  =  r 
ax^by  —  cz-\-du  =  s 

13.  2x  +  3y  —  4:Z  +  2u  +  3v  =  19 

dx-2y-\-2z  —  du  +  4:V  =  13 

2x  —  4:y-\-3z  —  2u  +  2v=    5 

ic+    y  —  dz-\-2u-\-    v=    7 

x-\-2y-{-dz  —  4:U-\-5v  =  2d 


818.  II.  To  determine  under  what  condition  (n  + 1) 
equations  of  n  unknown  quantities  may  he  simultaneously 
true. 

Assume  the  equations 

«i^  +  ^iy  +  ^i  =  0,  (A) 

«2^'  +  ^22/  +  ^2  =  0,  (B) 

and  «3  a^  +  ^3  y  +  ^3  =  0  (C) 

to  be  simultaneously  true. 

Multiply  (A)  by  d  ,  (B)  by  -  ^  ,  and  (C)  by  Cs ,  in  which  C^ , 
Ci ,  and  Oa  are  the  cofactors  of  ci ,  ca ,  and  Cs  in  the  determinant 
[  ai  62  C3  ],  and  add  the  results.    Then, 

(ai  C  —  aid  +  tti  Cs)  X  +  (bi  Ci  —  h  Ci  +  h  Cs)  y 

+  {ci  Ci  -  Ci  d  +  cs  Cz)  =  0  [Ax.  2]. 
But,  ai  C  —  aid  +  «3  G  =  &i  (7i  —  &2  Ci  +  bsC9  = 

ci  Ci  -Cid  +  cz  Cz  =  [  ai  bi  Cz]  [814,  P. ;  805,  R.]. 
.'.    [  «!  bi  C3  ]  =  0  is  the  condition  under  which  (A),  (B),  and 
(C)  are  simultaneously  true. 

Note. — The  equation  [  ai  bi  Ca  ]  =  0  is  called  the  eliminant  of 
the  group. 

In  a  similar  manner  it  may  be  shown  that  n  +  1  equations  of  n 
unknown  quantities,  of  the  form  of 

ai  a;  +  61  y  +  . . . .  +  ri  =  0, 
are  simultaneously  true,  when 

[ai    62   C8 rn4i  ]  =  0. 

Note.— Equations  that  are  simultaneously  true  are  said  to  consist^ 
that  is,  they  are  consistent. 


MULTIPLICATION  OF  DETERMINANTS,        44I 


EXERCISE     112. 

Test  the  consistency  of  : 
1.  2a;  +  3y-13  =  0 


2.  3a;-f-2y-17  =  0 
bx  —  '^y—  3  =  0 
2a;-5«/  +  15  =  0 


819.   III.   To  eliminate  x  from  any  two  rational  inte- 
gral equations  in  x. 

Ulustrations. — 1.  Eliminate  x  from  the  equations 


aoi?-\-  bx-\-c  =  0 
ma^-\-nx-{-r  =  0 
Solution :  It  is  evident  that 

ax^  +  bx^  +  ex        =  0 

ax^  +  bx  +  c  =  0 

mx^  +  nx^  +  rx         =0 


and 

mx'^  +  nx  +  r  =  0 

are  simultaneously  true. 

Therefore, 

a     b     c     0 

0     a     b     c 
m    n     r     0 

=  0 

0     m    n     r 

a 

b 

c 

d 

e 

0 

0 

a 

b 

c 

d 

e 

m 

n 

P 

<1 

0 

0 

0 

m 

n 

P 

? 

0 

0 

0 

m 

n 

P 

? 

(A) 
(B) 


2.  Eliminate  x  from  the  equations 

ax^-{-l)3^ -\- cx^ -\-dx-\- e  =0 
mQ?-\-noi?  -\-p  x-{-q=zO 
Solution :  It  is  evident  that 

ax^  +  bx!^  +  cx^  +  dx^ -{■  ex         =0 

ax!^  +  bx^  +  cx^  +  dx  +  e  =  0 

ma^ +  nx* +px^  +  qx^  =0 

mx^  +  nx^  +  px^  +  qx         =0 

ma^  +  nx^  -\- px  +  q  =  0 

are  simultaneously  true.    Therefore, 


=  0 


This  method  is  known  as  Sylvester's  Method  of  Elimination. 


442  ADVANCED  ALGEBRA, 

Probabilities. 

Definitions  and  Fundannental   Principles. 

820.  When  the  number  of  ways  in  which  an  event  may 
occur  is  greater  than  the  number  of  ways  in  which  it  may 
fail,  and  the  ways  are  equally  likely  to  happen,  we  say  : 

1,  The  event  is  'probable. 

2,  The  event  is  likely  to  happen, 

3,  The  chance  is  in  favor  of  the  event, 
U*  The  odds  are  in  favor  of  the  event, 

821.  When  the  number  of  ways  in  which  an  event  may 
fail  is  greater  than  the  number  of  ways  in  which  it  may 
occur,  and  the  ways  are  equally  likely  to  happen,  we  say  : 

1,  The  event  is  improbable, 

2,  The  event  is  not  likely  to  happen, 

3,  The  chance  is  against  the  event, 
U,  The  odds  are  against  the  event, 

822.  When  the  number  of  ways  in  which  an  event  may 
occur  is  equal  to  the  number  of  ways  in  which  it  may  fail, 
and  the  ways  are  equally  likely  to  happen,  we  say : 

1,  The  occurrence  and  failure  of  the  event  are  equally 
probable, 

2,  The  event  is  as  liJcely  to  happen  as  to  fail. 

3,  There  is  an  even  chance  for  and  against  the  event, 
U'  The  odds  are  even  for  and  against  the  event, 

823.  If  an  event  can  occur  in  a  ways  and  fail  in  b 
ways,  and  the  ways  are  equally  likely  to  happen,  we  need 
more  definite  language  to  express  the  exact  probability  or 
chance  of  the  event.     Thus,  we  say  : 

1,  The  odds  are  as  a  to  b  in  favor  of  the  event, 

2,  The  odds  are  as  b  to  a  against  the  event. 


PROBABILITIES.  443 

824.  If  we  let  k  represent  the  probability  of  any  par- 
ticular way  happening,  a  the  number  of  ways  favorable  to 
the  event,  and  h  the  number  of  unfavorable  ways,  then 
will  ah  represent  the  probability  of  the  event  happening, 
and  h  k  the  probability  of  its  failing,  and  ak-\-hk,  or 
{a  +  h)  k,  certainty^  which  is  taken  as  the  unit  of  measure, 

then  (a-\-V)k  =  \\  whence,  k  =         ,  ; 

and  ak  —         -, ,  the  probability  or  chance  of  the 

event,  and  Ik  —         , ,  the  probability  or  chance  against 

the  event.     Therefore, 

Prin,  1, — The  prolaMUty  or  chance  of  an  event  hap- 
pening equals  the  number  of  favorable  ways  divided  by  the 
whole  number  of  ways, 

Prin.  2. — The  probability  or  chance  of  an  event  fail- 
ing equals  the  number  of  unfavorable  ways  divided  by  the 
ivhole  number  of  ways, 

825.  Since  an  event  is  certain  to  happen  or  fail,  and 
certainty  is  expressed  by  unity,  it  follows  that, 

Prin,  3, — The  probability  of  an  event  happening  equals 
unity  minus  the  probability  that  it  will  fail  j  and  the  prob- 
ability that  it  will  fail  equals  unity  minus  the  probability 
that  it  will  happen. 

Illustration. — If  there  are  3  black  and  2  white  balls  in 
a  bag  containing  only  5  balls,  what  is  the  chance, 

1.  That  a  black  ball  will  be  drawn  on  the  first  trial  ? 

2.  That  a  black  ball  will  not  be  drawn  on  the  first  trial  ? 
Solution :  1.  There  are  3  favorable  ways  out  of  5  to  draw  a  black 

Q 

ball ;  therefore,  the  chance  is  -=•  (Prin.  1). 

3.  There  are  2  unfavorable  ways  out  of  5  to  draw  a  black  ball, 
namely,  the  two  favorable  ways  for  drawing  a  white  ball ;  therefore, 

2 
the  chance  of  failing  to  draw  a  black  ball  is  -g .    Or, 


444  ADVANCED  ALGEBRA, 

That  a  black  ball  will  be  drawn  or  not  drawn  on  the  first  trial 

Q 

is  certainty.    The  chance  for  drawing  a  black  ball  is  ■^;  therefore, 

3        2 

the  chance  of  failure  is  1  —  -^  =  -=■ . 

826.  Exclusive  Events. — Two  or  more  events  are  mu- 
tually exclusiye  when  the  happening  of  one  of  them  pre- 
cludes the  possibility  of  any  other  one  happening.  Thus, 
if  a  coin  be  thrown  up,  it  may  fall  either  head  or  tail.  If 
it  fall  head,  or  is  supposed  to  fall  head,  it  can  not  fall  tail, 
or  be  supposed  to  fall  tail,  in  the  same  throw.  Falling 
head  and  falling  tail  are,  therefore,  mutually  exclusive 
events. 

827.  In  a  bag  are  d  balls  ;  a  of  them  are  white,  h  blue, 
c  red,  and  the  remaining  ones  yellow.  What  is  the  chance 
of  drawing,  on  the  first  trial, 

1.  Either  a  red  or  a  white  ball  ? 

2.  A  red,  a  white,  or  a  blue  ball  ^ 

Solution :  1.  The  chance  of  drawing  a  red  ball  is  -r ,  and  the 
chance  of  drawing  a  white  ball  is  -^ ;  and  the  chance  of  drawing 

either  a  red  or  a  white  ball  is  — -j-  =  -j  +  -^ . 

2.  The  chance  of  drawing  a  red  ball  is  -^ ;  of  drawing  a  white 

h 

ball,  -^  ;  of  drawing  a  blue  ball,  -^  ;  and  of  drawing  a  red,  a  white,  or 
a  blue  ball, ^ ;  which  equals  "^  +  ^  +  "^  •    Therefore, 

JPrin.  4, — The  chance  that  one  of  several  mutually  ex- 
elusive  events  will  happen  equals  the  sum  of  their  separate 
chances  of  happening. 

EXERCISE     118. 

1.  "What  is  the  chance  of  throwing  4  with  a  single  die  ? 

Suggestion. — A  die  has  six  faces,  which  are  equally  liable  to  turn  up, 

but  only  one  of  these  contains  four  dots.    Therefore,  the  chance  is  -^  . 


PROBABILITIES.  445 

2.  What  is  the  chance  of  throwing  an  even  number 
with  a  single  die  ? 

, — Three  of  the  faces  have  an  even  number  of  dots; 


3  1 

therefore,  the  chance  is  ^ ,  or  ^. 

3.  If  the  odds  be  4  to  3  in  favor  of  an  event,  what  are 
the  respective  chances  of  the  success  and  failure  of  the 
event  ? 

Suggestion. — There  are  4  points  out  of  7  favorable  and  3  out  of  7 
unfavorable  to  the  happening  of  the  event ;  therefore,  the  respective 

chances  of  success  and  failure  are  -y  and  -=- . 

4.  If  4  coppers  are  tossed,  what  are  the  odds  against 
exactly  2  turning  up  head  ? 

Suggestion. — Each  coin  may  fall  in  two  ways;  hence,  the  four 
coins  may  fall  in  2*  =  16  ways  [550,  Cor.].    The  two  coins  that  may 

4x3 

turn  up  head  can  be  selected  from  the  four  coins  in     .^    ,  or  6  ways. 

fi  ^  — 

Therefore,  the  chance  of  success  is  ir^ ,  or  -5-  ?  and  the  chance  of  fail- 

3        5  16  8 

ure  is  1  —  -^  =  ^ .    Therefore,  the  odds  are  as  5  to  3  against  the 

event. 

5.  In  a  bag  are  7  white  and  5  red  balls ;  if  two  are 
drawn,  find  the  chance  that  1  is  red  and  1  white. 

12  X  11 
Solution :  Two  balls  can  be  selected  from  12  balls  in  — r^ —  =  66 

if. 
ways.    One  white  ball  can  be  selected  from  7  white  balls  in  7  ways,  and 
1  red  ball  from  5  red  balls  in  5  ways.    Hence,  1  white  ball  and  1  red 
ball  can  be  selected  from  7  white  and  5  red  balls  in  7  x  5,  or  35  ways. 
Therefore,  35  out  of  66  ways  are  favorable  to  drawing  1  white  and  1 

35 

red  ball.    Therefore,  the  chance  is  p^s . 

66 

6.  Twenty  persons  take  their  seats  at  a  round  table. 
What  are  the  odds  against  two  persons  thought  of  sitting 
together  ? 

Solution:  Let  the  two  persons  be  A  and  B.  Besides  the  place 
where  A  may  sit,  there  are  19  places,  two  of  which  are  adjacent  to 
him,  and  the  remaining  17  not  adjacent.  Any  of  these  B  may  select. 
Therefore,  the  odds  are  as  17  to  2  against  A  and  B  sitting  together. 


446  ADVANCED  ALGEBRA. 

828.  Expectation, — The  value  of  any  probability  of 
prize  or  property  depending  upon  the  occurrence  of  some 
uncertain  event  is  called  an  Expectation, 

7.  A  person  holds  a  tickets  in  a  lottery  in  which  the 
whole  number  of  tickets  issued  is  n.  There  is  only  one 
prize  offered,  and  this  is  worth  %p.  What  is  the  person's 
expectation  ? 

Solution :  It  is  evident  that  the  n  tickets  are  worth  ^p,  and  that 
the  tickets  are  of  equal  value  before  the  drawing;  therefore,  the  a 

tickets  are  worth  —  of  $  jo,  which  is  $  «  x  — .    Therefore, 

829.  Prin,  5. — The  expectation  of  an  event  equals  the 
product  of  the  sum  to  he  realized  and  the  chance  of  the 


8.  A  person  is  allowed  to  draw  two  bank-notes  from  a 
bag  containing  8  ten-dollar  bills  and  20  two-dollar  bills. 
What  is  his  expectation  ? 

28  X  27 
Solution :  The  two  notes  can  be  drawn  from  28  notes  in  — r^ — 

\jL 
=  378  ways.    Two  ten-dollar  notes  can  be  drawn  from  8  ten-dollar 

8x7 
notes  in     .        =  28  ways.    Two  two-dollar  notes  can  be  drawn  from 

—  20  X  19 

20  two-dollar  notes  in  — ^ —  =  190  ways. 

\A 
One  ten-dollar  note  and  one  two-dollar  note  can  be  drawn  from 
8  ten-dollar  notes  and  20  two-dollar  notes  in  8  x  20  =  160  ways. 
Therefore, 

28 
The  chance  of  drawing  $20  is  5^^,  and  the  expectation  is  $  1.482V* 

190 
The  chance  of  drawing  $4  is  -^^^ ,  and  the  expectation  is  $2.01-iV5. 

1  fiO 

The  chance  of  drawing  |12  is  5^=5 ,  and  the  expectation  is  $5.07ff . 

o7o 

.'.  The  entire  expectation  is  $1.48^  +  $2.01  ^  +  $5.07 1  =  $8.57  y. 

9.  A  bag  contains  a  £5  note,  a  £10  note,  and  six  pieces 
of  blank  paper  of  the  same  size  and  texture  as  a  bank-note. 
Show  that  the  expectation  of  a  man  who  is  allowed  to  draw 
out  one  piece  of  paper  is  £1  17^.  6d, 


PROBABILITIES.  447 

830.  Independent  Events, — Two  or  more  events  are  in- 
dependent of  each  other  when  the  happening  of  one  of 
them  does  not  affect  the  probability  of  any  other  one's 
happening. 

10.  There  are  h  balls  in  one  bag,  a  of  which  are  white  ; 
d  in  another,  c  of  which  are  white  ;  and  /  in  another,  e  of 
which  are  white.     Show  that  the  chance  of  drawing  one 

white  ball  from  each  bag  in  a  single  trial  is  -r  X  ^  X  ^ . 

Solution :  One  ball  can  be  drawn  from  each  bag  in  b  x  d  x  f  ways 
[550].    One  white  ball  can  be  drawn  from  each  bag  in  a  x  c  x  e  ways 
[550].    Therefore,  the  chance  of  drawing  a  white  ball  from  each  bag 
.    a  X  c  X  e  ^^..  T^  ^^        a       c        e       _,,       . 
■«  hlTdlTf  t^^*'  P- 1]  =  6  "^  d  ><  7 •    T'^^'^'o^^' 

831.  JPrin.  6, — The  chance  of  two  or  more  independent 
events  happening  simultaneously  is  the  product  of  their 
several  chances  of  happening, 

832.  Car.  1, —  The  chance  of  two  or  more  independent 
events  failing  simultaneously  is  the  product  of  their  several 
chances  of  failing, 

833.  Cor,  2. — The  chance  of  one  of  two  independent 
events  failing  and  the  other  happening  is  the  product  of 
the  chance  that  one  fails  and  the  chance  that  the  other 
happens. 

11.  A  can  solve  3  problems  out  of  4,  B  5  out  of  6, 
and  C  7  out  of  8.  What  is  the  chance  that  a  certain 
problem  will  be  solved,  if  all  try  ? 

Solution :  Unless  all  fail,  the  problem  will  be  solved.    The  chance 

that  A  will  fail  is  i ,  that  B  wUl  fail  -^ ,  that  C  wiU  fail  g  ,  that  aU 
will  fail  -J  X  -^  X  -^  =  Yq^  •    Therefore,  the  chance  of  success  is  -^ , 

834.  Dependent  Events. — In  a  series  of  events,  any 
assumed  event  is  said  to  be  dependent  upon  a  preceding 


4AS  ADVANCED  ALGEBRA. 

event,  if  the  happening  of  the  preceding  event  changes  the 
probability  of  the  happening  of  the  assumed  event. 

12.  Find  the  chance  of  drawing  3  white  balls  in  suc- 
cession from  a  bag  containing  5  white  and  3  red  balls. 

Solution :  The  chance  of  drawing  a  white  ball  on  the  first  trial 

K 

is  -5- .    Having  drawn  a  white  ball,  there  remain  in  the  bag  7  balls, 

o 

4  of  which  are  white.    The  chance  of  drawing  a  white  ball  on  the 

4 
second  trial  is  therefore  -=- .    Similarly,  the  chance  of  drawing  a  white 

Q 

ball  on  the  third  trial  is  -k  .    Therefore,  the  chance  of  drawing  three 

5       4       3  5 

white  balls  in  succession  is  q-  x  -=-  x  -^  [831]  =  -^r^ .    Therefore, 

835.  Prin,  7.  —  The  chance  that  a  series  of  events 
should  happen  is  the  continued  product  of  the  chance  that 
the  first  should  happen,  the  chance  that  the  second  should 
then  happen,  the  chance  that  the  third  should  follow,  and 
so  on, 

13.  In  one  of  two  bags  are  3  red  and  4  white  balls,  and 
in  the  other  5  red  and  3  white  balls,  and  a  ball  is  to  be 
drawn  from  one  or  other  of  the  bags.  Find  the  chance 
that  the  ball  drawn  will  be  white. 

Solution :  The  chance  that  the  first  bag  will  be  chosen  is  -^ .    Then, 

a 

4 
the  chance  of  drawing  a  white  ball  from  the  first  bag  is  -y ;  hence,  the 

14       2 
real  chance  of  drawing  a  white  ball  from  the  first  bag  is  -^  of  -=r  =  -=^ . 

Similarly,  the  chance  of  drawing  a  white  ball  from  the  second  bag  is 

13        3 

—  of  -Q  =  T5  •    These  events  are  mutually  exclusive ;  therefore,  the 

\       ^       ^  .     3       3        53 
chance  required  ^^  -y  +  jg  =  htr  • 


836.  Inverse  Probability. — When  an  event  is  known 
to  have  happened  from  one  of  two  or  more  known  causes, 
the  determination  of  the  chance  that  it  has  happened  from 


PROBABILITIES.  449 

any  particular  one  of  these  causes  is  a  problem  of  inverse 
prohalility, 

14.  It  is  known  that  a  black  ball  has  been  drawn  from 
one  of  two  bags.  The  first  of  these  bags  contained  in  balls, 
a  of  which  were  black,  and  the  second  n  balls,  b  of  which 
were  black.  What  is  the  chance  that  the  ball  was  drawn 
from  the  first  bag  ? 

Solution :  Suppose  that  2  N  drawings  were  made.  The  chance  is 
that  N  were  made  from  each  bag.    In  the  N  drawings  from  the  first 

bag  the  chance  is  that  ~  x  N  were  black  balls.    In  the  drawings 

from  the  second  bag  the  chance  is  that  —  x  iV  were  black  balls. 

Therefore,  in  2  JV  drawings,  the  chance  is  that  (  —  H jN  were 

black  balls.    Therefore,  the  chance  that  a  black  ball  was  drawn  from 

the  first  bag  is  (—  xiV")  -^(—  +  —)  N  =  — "'\     . 

837.  Theorem, — If  an  event  is  believed  to  have  been 

produced  by  some  one  of  the  causes  Pi,  Pg?  A^ P»> 

which  are  mutually  exclusive,  and  Pi,  Pzy  Pzy Pn  rep- 
resent the  respective  probabilities  of  these  causes  when  no 
other  causes  exist,  then  the  probability  that  P,  produced 

the  event  is  -^ t—^ "-p^ i-^ —  . 

Demonstration. — Let  N  be  the  number  of  trials  made  in  produc- 
ing the  event.  The  first  cause  operated  N  x  Pi  times ;  therefore, 
on  the  supposition  that  no  other  causes  operated  than  those  named, 
the  probability  that  the  event  was  produced  by  the  first  cause  is 
iV  X  Pi  X  pi.  Under  similar  restrictions,  the  probability  that  the 
event  was  produced  by  the  second  cause  is  iV  x  Pg  x  jsa ;  by  the  third 
cause,  N  X  Pz  x  ps;  by  the  rth  cause,  N  x  Pr  x  pr  ;  by  any  one  of 

the  causes,  N{Pipi  +  P^p^  +  Psps  + +  P«i?«).     Therefore,  the 

real  chance  of  its  having  been  caused  by  the  rth  cause,  or  P, ,  is 

N  X  Pr    X  Pr  _  Pr    X  Pr 

N{Pipi   +  Ps^s  +....  +  PnPn)  ~  P\  P\    +  P^Pi   + +  PnPn' 

15.  Four  bags  were  known  to  contain  3  red  and  4  white, 
4  red  and  3  white,  5  red  and  1  white,  and  4  red  and  4 


450  ADVANCED  ALGEBRA. 

white  balls  respectively.  A  white  ball  was  drawn  at  ran- 
dom from  one  of  the  bags.  Find  the  chance  that  it  was 
drawn  from  the  second  bag. 

Solution:   Pi  =P2  =  Ps  =  P4  =  i,  i?i  =  y,  i>2  =  |,   Pi  =  \, 
and  i?4  =  o" .    Therefore,  the  required  probability  is 

1      ^  i 

4^7  T        9 


2_/4       i       1       1\       :5 
4V7  "^  7  "*■  6  ■*■  2/       3 


35 


838.  Probability  of  Testimony.— HhQ  following  exam- 
ples illustrate  how  to  deal  with  questions  relating  to  the 
credibility  of  testimony : 

16.  A  speaks  the  truth  a  times  in  W2,  B  J  times  in  n, 
and  0  c  times  in  r.  What  is  the  chance  that  a  statement 
is  true  which  all  affirm  ?  Which  A  and  B  affirm  and 
C  denies  ? 

Solution :  1.  The  statement  is  either  true  or  false.    If  true,  all  have 

spoken  the  truth ;  the  probability  of  which  is  —  x  —  x  —  = . 

^  '         ^  •'  m       n       r       mnr 

If  false,  all  have  lied ;  the  probability  of  which  is 

\        m)\         n)\         r)  mnr 

Hence,  the  probability  of  the  truth  of  the  statement  is, 
ahc    ^    {abc       (m—a){n—b)(r—c))_  abc 

mnr  '    {mnr  mrir  )   ~  abc  +  {m—a){n—b)(r—c)' 

2.  If  the  statement  is  true,  A  and  B  have  told  the  truth  and  C 

has  lied ;  the  probability  of  which  is  —  x  —  xfl )  =  — ^^ , 

^  ■'  m       n       \         r  J         mnr 

If  the  statement  is  false,  A  and  B  have  lied  and  C  has  told  the  truth ; 

the  probability  of  which  is  fl-  »)  (l-  *  )  (^)  =  '"'-"»"-^>'. 
^  J  \       mj\        nj\rj  mnr 

Hence,  the  probability  of  the  truth  of  the  statement  is, 

ab{r—c)  ^    \ab{r—c)      {m—a){n—b)c\_ a b (r—c) 

mnr      '    {    mnr  mnr        )  ~  ab{r—c)  +  {m—a){n—b)c 

17.  A,  B,  and  0  tell  the  truth  to  the  best  of  their 
knowledge  and  belief.  A  observes  correctly  4  times  out 
of  5,  B  3  times  out  of  5,  and  C  5  times  out  of  7.  What 
is  the  probability  that  a  phenomenon  occurred  (which  was 


PROBABILITIES.  451 

just  as  likely  to  fail  as  to  occur),  provided  all  had  equal 
opportunity  of  observing,  and  all  report  its  occurrence  ? 
What  if  A  and  B  report  its  occurrence  and  0  its  failure  ? 
Solution :  1.  The  phenomenon  either  occurred  or  failed.  If  it  oc- 
curred, A,  B,  and  C  observed  correctly ;  the  probability  of  which  is 

-^  X  -^  X  -=-.    The  inherent  probability  that  it  would  occur  is  -^ . 
o        o        7  -© 

Hence,  the  probability  that  the  assumption  that  it  occurred  is  correct 
•    1      1      1      A -A 

is  2  X  g  X  g  X  ^         _gg. 

If  it  did  not  occur,  all  observed  falsely ;  the  probability  of  which 

12        2 

is  -^  X  -=■  X  ■=■ ;  and  the  probability  of  the  correctness  of  the  assump- 

11222 
tion  that  the  phenomenon  failed  is7rX-=-x-=-x-=-  =  r-— ,   Hence, 

2        0        5        7        175 

the  chance  that  the  phenomenon  occurred  is  5^  -^  (  ^^  +  7^  )  =  -- . 

do       \35       175/       16 

2.  The  probability  of  the  correctness  of  the  assumption  that  the 

,  .     1        4        3        2        12 
phenomenon  occurred  iS7rX-=-x-=-x-7=-  =  77==. 
^  2        5        5        7       175 

The  probability  of  the  correctness  of  the  assumption  that  the 

,  ..,,.1125        1 

phenomenon  failed  \s  -z  -k  -^  y.  -^  y.  -p^  =  ^. 

Hence,  the  chance  of  the  event  is  ^^  -«-  f  r^  "^  s^)  ~  17* 

Note. — For  a  fuller  treatment  of  Choice  and  Chance  than  space  will 
permit  to  give  in  this  book,  see  Whitworth's  "  Choice  and  Chance." 


EXERCISE     114. 

1.  If  A's  chance  of  winning  a  race  is  —  and  B's  chance 

1  17 

■Z-.  show  that  the  chance  that  both  will  fail  is  rr. 
o  24 

2.  If  the  odds  be  m  to  w  in  favor  of  an  event,  show 

that  the  chance  of  the  event  is  — ; — ,  and  the  chance 

m-{-n 

against  the  event  is  — ; — . 
7n-\-n 

3.  If  the  letters  e,  t,  s,  n  be  arranged  in  a  row  at  ran- 
dom, show  that  the  chance  of  having  an  English  word  is  — . 

M  O 


452  ADVANCED  ALGEBRA. 

4.  Show  that  the  chance  that  the  year  1900  +  2?,  in 
which  X  <  100,  is  a  leap-year,  is  ^ . 

5.  A  draws  3  balls  from  a  bag  containing  3  white  and 
6  black  balls  ;  B  draws  1  ball  from  another  bag  containing 
1  white  and  2  black  balls.  Show  that  A's  chance  of  draw- 
ing a  white  ball  is  to  B's  chance  as  16  to  7. 

6.  Show  that  when  two  dice  are  thrown  the  chance  that 
the  throw  will  amount  to  more  than  8  is  -5 . 

lo 

7.  Show  that  the  chance  of  throwing  exactly  11  in  one 
throw  with  two  dice  is  :7^ . 

lo 

8.  One  purse  contains  5  sovereigns  and  4  shillings ; 
another  contains  5  sovereigns  and  3  shillings.     Show  that 

the  chance  of  drawing  a  sovereign  is  —rj ,  if  a  purse  is 

selected  at  random  and  a  coin  drawn  from  it  at  random. 
Show  that  the  expectation  of  the  privilege  is  125.  2  Yig  d, 

9.  There  are  three  independent  events  whose  several 
chances  are  ^,  -r-,  and  ■^.  Show  that  the  chance  that 
one  of  them  will  happen  and  only  one  is  j^ . 

10.  If  two  letters  are  taken  at  random  out  of  esteemed, 
show  that  the  odds  against  both  being  e  are  the  same  as 
the  odds  in  favor  of  one  at  least  being  e, 

y.     11.  A  letter  is  taken  at  random  out  of  each  of  the 
words  choice  and  chance.     Show  that  the  chance  that  they 

are  the  same  letter  is  — . 
b 

12.  A  bag  contains  6  black  and  1  red  ball.  Show  that 
the  expectation  of  a  person  who  is  to  receive  a  shilling  for 
every  ball  he  draws  out  before  drawing  the  red  one  is  3 
shillings. 


PROBABILITIES.  453 

13.  Two  numbers  are  chosen  at  random.  Show  that 
the  chance  is  ^  that  their  sum  is  even. 

14.  An  archer  hits  his  target  on  an  average  3  times  out 

of  4.     Show  that  the  chance  that  he  will  hit  it  exactly 

27 
3  times  m  4  successive  trials  is  77-7 . 

15.  A  box  contains  10  pairs  of  gloves.     A  draws  out  a 

single  glove  ;  then  B  draws  one  ;  then  A  draws  a  second  ; 

then  B  draws  a  second.     Show  that  A's  chance  of  drawing 

a  pair  is  the  same  as  B's  ;  and  that  the  chance  of  neither 

^       .  .    .    290 

drawmg  a  pair  is  ^^ . 

16.  Show  that  with  two  dice  the  chance  of  throwing 
more  than  7  is  equal  to  the  chance  of  throwing  less  than  7. 

17.  Two  persons  throw  a  die  alternately,  with  the  under- 
standing that  the  first  who  throws  6  is  to  receive  11  cents. 
Show  that  the  expectation  of  the  first  is  to  that  of  the 
second  as  6  to  5. 

18.  A's  chance  of  winning  a  single  game  against  B  is  ^ . 

Show  that  his  chance  of  winning  at  least  2  games  out  of  3 

.     81 

^^-25- 

19.  A  party  of  n  persons  take  their  seats  at  random  at 
a  round  table.  Show  that  it  is  w  —  3  to  2  against  two 
specified  persons  sitting  together. 

20.  Show  that  the  chance  that  a  person  with  2  dice 
will  throw  double  aces  exactly  3    times  in    5    trials  is 


\m)  ^  \36J 


X  10. 


21.  There  are  10  tickets,  five  of  which  are  numbered 
1,  2,  3,  4,  5,  and  the  rest  are  blank.  Show  that  the  prob- 
ability of  drawing  a  total  of  ten  in  three  trials,  one  ticket 

33 

being  drawn  each  time  and  replaced,  is  r^r^ . 


SUPPLEMEl^T. 


COJ^TIJfUED    FBACTIOJfS, 


I.   Definitions. 
839.  An  expression  in  the  form  of 


a 


h-{-c 


d-\-e 


/+  etc., 
is  a  Continued  Fraction, 

840.  The  discussion  in  this  section  will  be  limited  to 
continued  fractions  in  the  form  of 

1 

«4-i 


J+i 


c+  etc., 

and  -,  -7-,  -,  etc.,  will  be  called  Partial  Fractions, 
a'  h     c 

841.  A  continued  fraction  may  be  written  in  a  more 
convenient  form,  as  follows  : 

111  ill 

842.  When  the  number  of  partial  fractions  in  a  con- 
tinued fraction  is  finite,  it  is  a  terminating  continued  frac- 
tion ;  when  infinite,  an  interminate  continued  fraction. 

843.  If  at  some  stage  in  an  interminate  continued  frac- 
tion one  or  more  partial  fractions  begin  to  repeat  in  the 
same  order,  it  is  called  a  periodic  continued  fraction. 


CONTINUED  FRACTIONS.  455 

844.  A  periodic  continued  fraction  is  pure  when  it 
contains  no  other  than  repeating  partial  fractions,  and 
mixed  when  it  contains  one  or  more  partial  fractions  be- 
fore the  repeating  ones. 


rrt.         1        1        1        1 

1       1 

is  a  pure  periodic  fraction ; 

11111 

1      i      i 

is  a  mixed  periodic  fraction. 

845.  The  fraction  resulting  from  stopping  at  any  stage 
is  called  a  convergent, 

2.  The  Formative  Law  of  Successive  Convergents. 

846.  In  the  continued  fraction 
111  1      1      1      i 

—  =  the  first  convergent. 
-   ,   -T  =     r  ,  -.  =  the  second  convergent. 

-   ,   -T   ,   -  =  .    ,   ,  ..x — i —  =  the  third  convergent. 
a  -{■  b  -{-  c       {ab-\-l)c-{-a  ^ 

It  will  be  seen  that 

t  The  numerator  of  the  third  convergent  is  the  numer- 
ator of  the  second  convergent  multiplied  by  the  denomina- 
tor of  the  third  partial  fraction,  plus  the  numerator  of  the 
first  convergent;  and 

2.  The  denominator  of  the  third  convergent  is  the  de- 
nominator of  the  second  convergent  multiplied  by  the  de- 
nominator of  the  third  partial  fraction,  plus  the  denomi- 
nator of  the  first  convergent. 


456  ADVANCED  ALGEBRA. 

Will  these  laws  hold  true  in  the  formation  of  any  con- 
vergent from  the  two  preceding  convergents  ? 

P       O       Tf  ^ 

Let  p-,  -~f  -^,  and  -^  be  respectively  the  (/i  — 2)th, 
^1      Vi     -^1  ^1 

{n  —  l)th,  nth,  and  {n  +  l)th  convergents ;  and  p,  q,  r, 
and  s  the  denominators  of  the  (n  —  2)th,  {n  —  l)th,  nth, 
and  (?i  +  l)th  partial  fractions. 

Suppose  the  laws  to  hold  true  in  the  formation  of  the 

convergent  ^^ ,  then  will  ^  =  7Q^,  '  (^) 

Now,  from  the  nature  of  the  continued  fraction,  -^  may 
1  R  ^1 

be  formed  by  putting  r-\ —  f or  r  in  -^ .     Therefore, 

S  2il 


^_(r+l)8+i- 


(«r  +  l)e  +^ 

?P 

(sr  +  l)e.  +  ^ 

sA 

sB  +Q 

_^(re+p)+e 

Therefore,  if  the  laws  are  applicable  in  the  formation 
of  the  nth.  convergent,  they  are  also  applicable  in  the 
formation  of  the  {n  +  l)th  convergent.  But  we  have 
seen  that  they  do  apply  in  the  formation  of  the  third 
convergent,  and,  hence,  apply  in  the  formation  of  the 
fourth  convergent,  and  so  on.     Therefore,  in  general, 

1.  The  numerator  of  the  nth  convergent  equals  the 
numerator  of  the  (n  —  l)th  convergent  multiplied  by  the 
denominator  of  the  nth  partial  fraction,  plus  the  numera- 
tor of  the  {n  —  2)th  convergent ;  and 

2.  The  denominator  of  the  nth  convergent  equals  the 
denominator  of  the  (n  —  l)th  convergent  multiplied  by  the 
denominator  of  the  nth  partial  fraction,  plus  the  denomi- 
nator of  the  (n  —  %)th  convergent. 


PROPERTIES  OF  CONVEROENTS,  457 

Example. — Find  the  first   8   convergents  of  the  con- 

tmued  fraction  ^^3_^i^4^5^^  +  4  +  3- 
Solution : 

R  _rQ   -\-P  _\     i     i     1?     ^     ?1?     ^     3118 
Ri~  rQx-^Px~  2'    7'    9'  43'  324'  491'  2188'  7055" 


Properties  of  Convergents. 

847.  Take  the  continued  fraction 
_1       1       1       1 

«zx.ll  .1>..11 

whence,  i^i<2/ 

I,  .1       1^.111 

whence,  «+ -  _^  -  <  «  +  -_^-^-_^.. . . ; 

and  -   ,   -T   ,   -  >  Vi  etc.     Therefore, 

Prin.  1, — The  successive  convergents  are  alternately 
greater  and  less  than  the  continued  fraction  {the  odd  orders 
being  too  great  and  the  even  orders  too  small). 


848.  The  difference  between  the  first  two  conyergents 

= 5— 7-T  =     ,   ,   .  ^x  =  unity  divided  by  the  prod- 

a       aJ  +  1       a  (a  5  +  1)  ^  ^       ^ 

uct  of  their  denominators.     Is  this  a  general  law  ? 

P       Q  J? 

Let  -^f  -TT,  and  ^-  be  the  (w— l)th,  wth,  and  {n-\-l)th. 
convergents,  and  p,  q,  and  r  the  denominators  of  the 


458  ADVANCED  ALGEBRA. 

{n  —  l)th,  nth,  and  {n  -f  l)th  partial  fractions,  and  let  '^ 
denote  difference  hetioeen. 

Assume  -—  r^ -^  =  —  „  ^ — -  =  „   ..  ;  then  will 

Ri'^  Qi  QiRi  QiRi 

_  PQ.^QPr  _  _j_ 

"      Gi^i      "ft  A*  ^^ 

Therefore,  if  the  law  holds  good  for  the  difference  be- 
tween the  {n  —  l)th  and  wth  convergents,  it  will  also  for 
the  difference  between  the  {n  +  l)th  and  the  wth  conver- 
gents. But  we  have  seen  that  it  does  hold  good  for  the 
difference  between  the  first  and  second  convergents,  and, 
hence,  it  will  for  the  difference  between  the  next  higher 
pair,  and  so  on.     Therefore, 

JPrin.  2, — The  difference  between  any  two  consecutive 
convergents  equals  unity  divided  by  the  product  of  their 
denominators.  

849.  Since  PQ^'^QP^  =  1  [848,  A],  P  and  P^  can 
not  have  a  common  factor,  neither  can  Q  and  Q^. 
Therefore, 
Prin,  3, — Bvery  convergent  is  in  its  lowest  terms. 


850.  If  we  let  ^=-  represent  the  true  value  of  the  con- 
tinned  fraction ;  then  will 

P         U   ^P        Cmn       1        ^        Q   ^  P        Q 

'''  p[-u.<p;Q.^^'''^''''''urQ;^p;Qr' 

Hence,  if  either  -^  or  ^  be  used  for  -^,  the  error  will 
be  less  than  p  ^  ,  or  less  than  -^ . 

-^1  Vi  VI 


PROPERTIES  OF  CONVERGENTS,  459 

P       Q  R 

851.  Let  -^,  -jr,  and  -^  be  three  consecutive  con- 

-t  1     vi  ^1  -^     ^ 

versrents  whose  terminal  partial  fractions  are  -,  -,  and 
1  U  ,      P     ^ 

—  :  and  -77- ,  the  true  value  of  the  continued  fraction. 
r  Ui 

TT  7?  1 

Then,  y^  differs  from  ^-  only  in  the  use  of  r  H — —  etc. 
t/i  ^        III  s-\-       , 

for  r.     Put  r  H r  etc.  =  a;. 

s-\r 

1 


ftC^ft  +  i'i) 


,     P        U  _P_       xQ  +P  _x{PQ^-P,Q) 


a; 


A  (a:  ft  +  Pi) 
Now,  a;  >  1,  and  Pi  <  ft , 

1  ^  a; 

<  p  /^  /.    .    D  X  ;  or. 


••  ft(^ft  +  i^i)  ^A(^ft  +  A) 

-^  is  nearer  -^  than  is  ^-.     Therefore, 
vi  ^1  -^1 

2*Hn.  4. — The  MgJier  the  order  of  a  convergent  the 

nearer  does  it  approach  to  the  true  value  of  the  continued 

fraction. 

852.  Cor. — A  continued  fraction  is  the  limit  of  its  con- 
vergent s  ;  or,  if  y  be  a  continued  fraction  and  x  its  vari- 
able convergent,  y  =  lim.  x. 


853.  The  denominators  of  successive  convergents  in- 
crease more  rapidly  than  their  numerators  [846 ;  1,  2] ; 
therefore,  of  any  two  convergents,  that  is  the  greater 
which  has  the  greater  denominator.     But  may  there  not 


460  ADVANCED  ALGEBRA. 

be  some  other  fraction,  not  a  convergent,  with  smaller  de- 
denominator,  that  is  a  nearer  approximation  to  a  continued 
fraction  than  a  given  convergent  ? 

Suppose  -Yjr  not  a  convergent,  and  nearer  to  -^  than  ■—, 

^^  U       M        Q        U         ^'  ^' 

and  ifi  <  §1 ;  then  —  ^  -^  <  ^  ^  -^ ; 

M       P    ^  Q       P  MP^^M^P 


/^ 


<  -^  --  D- ;  or, TT-B < 


But  MtPi  <  QiPi,  since  M^  <  ft. 
,'.  MPi  '^  MiP  <  1;  which  is  impossible,  since  M, 
Ml,  P,  and  Pi  are  integral.     Therefore, 

Brin*  5, — Any  convergent  is  nearer  the  true  value  of 
a  continued  fraction  than  any  fraction  with  smaller  de- 
nominator. 

Problems. 

851.    1.   To  reduce  a  common  fraction  to  a  terminating 
continued  fraction. 

Since  an  improper  fraction  is  equivalent  to  an  integer 
and  a  proper  fraction,  it  will  be  necessary  only  to  investi- 
gate a  method  for  extending  a  proper  fraction. 

Let  -  =  a  proper  fraction  in  its  lowest  terms. 

Divide  both  terms  by  h,  and  put  for  the  improper  frac- 

a  c 

tion  -T ,  the  mixed  number  p-\-  -r]  then. 


h 
a 

1 

c 
1 

Divide  both  terms  of  -r 

0 

by  c,  and 

put 

h  _ 
c 

=  ^  + 

d 

n,                      J 
a 

1 

1 

q^d 
c 

PROPERTIES  OF  CONVERGENTS.  461 

d  c  p 

Again,  divide  both  terms  of  -  by  d,  and  put  -7  =  r  +  77  ; 

then,  _5  ^  1       1       1 

a  ~  p  -\-  q  -\-  r  -\-  e_ 
d 
It  will  now  be  seen  that  the  denominators  of  the  suc- 
cessive partial  fractions  have  been  obtained  as  follows : 
b)  a  {p 
bp 

11. 
d)  c  (r 
rd 
e  etc. 
Since  a  and  I  are  integral,  they  have  a  highest  com- 
mon divisor,  and  the  division  will  eventually  terminate. 

Therefore,  the  continued  fraction  will  be  a  terminat- 
ing one, 

Rule. — To  reduce  a  proper  fraction  to  a  terminating 
continued  fraction,  find  the  highest  common  divisor  of  its 
terms  by  successive  division,  and  use  the  quotients  in  regu- 
lar order  for  the  denominators  of  the  partial  fractions, 

855.    2.    To   reduce   a   quadratic   surd  to  a   continued 
fraction. 

Ulnstrations. — 1.  Eeduce  ^26  to  a  continued  fraction. 

Solution :  V36  =  5  +  — 

=  10  +  :4—    .     =  10  +     "^ 


10  +    1  ^   10  +     1 

10  + Jl_ 

X 


■•       '»^  =  «+fo  +  K  +  ^+--=5+^ 


462  ADVANCED  ALGEBRA. 

2.  Eeduce  a/iQ^  to  a  continued  fraction. 
Solution :  ^19  =  4  +  — 


X 

Vl9-4  3  ccj 

3             Vl9  +  2       ,       1 
iCi  =  —y= =  - — = =  1  +  — 

5  ^19"+ 3       „       1 

ars  =  -7= =  - — s =  3  +  - 

VlQ  -  3  2  iCs 

Scholiwm, — A  quadratic  surd  may  always  le  reduced 
to  a  periodic  continued  fraction  if  the  expansion  is  carried 
sufficiently  far, 

856.    3.    To  reduce  a  periodic  continued  fraction  to  a 
simple  fraction. 

The  periodic  continued  fraction 

ili_lll  __ 

p  +  q  -h  r  ~  p-\-  q  -i-  r  -\-x~ 
qr  +  qx-^1 _^ 


pqr-\-pqx-\-p-{-r-\-x 
whence,  {pq-{-l)x^-\-ipqr  —  q -\-p  +  r)x  =  qr-^-l 

The  value  of  x  found  from  this  equation  is  the  value 
of  the  continued  fraction. 

857.    4.  To  approximate  the  ratio  of  two  numbers. 

Example. — When  the  diameter  of  a  circle  is  1,  the  cir- 
cumference is  3  •1415926  +  .  Approximate  the  ratio  of  the 
diameter  to  the  circumference. 

Solution : 

-I     oi^iKno«       10000000       1111  rp^  ,    ., 

1 : 3.1415926  =  3^jj^=  3  ^  y  ^  j^  ^  J  ^....  [Prob.  1]. 

The  successive  convergents,  which  are  also  the  successive  approxi- 

^  ^,        ^.  17     106     113      , 

mationsof  the  ratio,  are :  -g,  ^,  ^,  g^g,  etc. 


PROPERTIES  OF  CONVERGENTS,  46_3 

EXERCISE     1 18. 

Reduce  to  continued  fractions  : 


125 

^'  317 

140                     100                     106 
^'  213                ^"  999               ^'  729 

6.  ViO 

6.  Vl2              7.  a/30              8.  V57 

9.  -3183 

10.  3-1416        11.  67°,  20',  30" 

Find  the  successive  convergents  of  : 

"•UJ  + 

^        ^                13    ^        ^        ^        ^ 

4+5                      3+1+9+1 

i      i 

"■2  +  3 

15     ^          ^          ^                 16     ^          ^ 

Find  the  true  value  of  : 

17.    -r              18. 

4 

i      i      _Q  1      i      „  i      1      i 

2  +  4        *^"  2  +  4        ^"-1  +  2  +  1 

21.  Find  a  series  of  common  fractions  converging  to 
1:  y/J. 

22.  Express  approximately  the  ratio  of  a  liquid  quart 
(57*75  cu.  in.)  to  a  dry  quart  (67*2  cu.  in.). 

23.  The  square  root  of  600  is  24*494897,  and  the  cube 
root  of  600  is  8*434327.  Find  a  series  of  four  common 
fractions  approximating  nearer  and  nearer  to  the  ratio  of 
the  latter  to  the  former. 

24.  The  imperial  bushel  of  Great  Britain  contains 
2218*192  cu.  in.,  and  the  Winchester  bushel  2150*42  cu. 
in.  Find  the  nearest  approximation,  that  can  be  expressed 
by  a  common  fraction  whose  denominator  is  less  than  100, 
of  the  ratio  of  the  latter  to  the  former. 

25.  Two  scales  of  equal  length  having  their  zero  points 
coinciding  also  have  the  27th  gradation  of  the  one  to  coin- 
cide with  the  85th  gradation  of  the  other.  Show  that  the 
7th  and  22d  more  nearly  coincide  than  any  other  two  gra- 
dations. 


464  ADVANCED  ALGEBRA. 

THEORY    OF   JfUMBEBS. 


Systems  of  Notation. 

I.   Definitions. 

858.  Notation  is  the  art  of  expressing  numbers  by 
means  of  characters. 

869.  A  system  of  notation  is  a  method  of  expressing 
numbers  in  a  series  of  powers  of  some  fixed  number. 

860.  The  order  of  progression  on  which  any  system  of 
notation  is  founded  is  called  the  scale  of  the  system,  and 
the  fixed  number  on  which  the  scale  is  based  is  called  the 
radix, 

861.  Any  integral  number,  except  unity,  may  be  taken 
as  the  radix.  When  the  radix  is  two,  the  scale  and  system 
are  called  binary ;  when  three,  ternary ;  when  four,  qua- 
ternary ;  when  five,  quinary ;  when  six,  senary ;  when 
seven,  septenary  ;  when  eight,  octary  ;  when  nine,  nonary ; 
when  ten,  denary  or  decimal ;  when  eleven,  undenary ; 
when  twelve,  duodenary ;  etc. 

862.  In  the  decimal  or  denary  system, 

56342  =  5x10,000  +  6x1000  +  3x100  +  4x10  +  2  = 
5Xl0*  +  6xl03  +  3Xl02  +  4xl0  +  2;  , 

or,  in  inverse  order, 

2  +  4x10  +  3x102  +  6x10^  +  5x10*. 

In  the  octary  system, 

34725  =  3  X8*  +  4X83  + 7x88  +  2x8  +  5; 
or,  in  inverse  order, 

5  +  2x8  +  7x82  +  4x83  +  3x8*. 


SYSTEMS  OF  NOTATION.  465 

.*.  In  general,  if  r  be  taken  as  the  radix,  and 

ttQ,    a-i,    «3,    «3  .  .  .  .  ««_! 

as  the  n  digits  of  a  number,  reckoning  in  order  from  right 

to  left,  the  number  is  represented  by 

«•-!  y-""^  +  a^«-2  ^""^  +  ««-3  ^""^  +  ....  +  «2  r^  -}-  «i  r  +  «o 

863.  Theorem, — Any  integral  number  may  he  expressed 
in  the  form  of 

ar^'-^-h  r"~^  +  ^  ^*~^  + -\-p  r^-\-qr  +  s, 

in  which  the  coefficients  are  each  less  than  r. 

Demonstration :  Let  N  equal  the  number  of  units  in  any  number, 
and  r"  the  highest  power  of  the  radix  less  than  N. 

Divide  N  by  r",  and  let  the  quotient  be  a  and  the  remainder  N', 
Then  N=ar''  +  N'. 

Now,  a  is  less  than  r,  else  r"  would  not  be  the  highest  power  of 
r  less  than  N;  and  JV'  is  less  than  r". 

Divide  N'  by  r«-^  and  let  the  quotient  be  b  and  the  remainder 
iV".    Then  N'  =  Ir^-"^  +  iV",  in  which  6  <  r  and  N"  <  r«-i. 

In  like  manner,  divide  N"  by  r*-^,  and  let  the  quotient  be  c,  and 
the  remainder  N'".  Then  iV"  =  cr'-^  +  N"\  in  which  c  <r  and 
N'"  <r*-8. 

If  this  process  be  continued,  a  remainder,  5,  will  eventually  be 

reached  less  than  r.     Therefore,  JV=  ar*  +  6r*— *  +  cj*—''^+ 

+  pr^  +  qr  +  s,  in  which  the  coeflacients  are  each  less  than  r. 

864.  Car, — In  any  system  of  notation,  the  number  of 
digits  including  0  is  equal  to  the  radix, 

866.    Problem.    To  express  a  given  number  in  any  pro- 
posed scale. 

Solution :  Let  N  be  the  number  and  r  the  radix  of  the  proposed 
scale. 

Suppose    JV=  ar"  +  &r«-»  +  cr«-2  + +  pr^  +  qr  +  s,  it  is 

required  to  find  the  values  of  a,  b,  c, p,  q,  s. 

N  8 

—  =  ar^-^  +  br^-^  +  ci-*^—^  + +  pr  +  q  ■{ — . 

r  T 

Therefore,  the  remainder,  after  dividing  N  by  r,  is  the  last  digit. 

Suppose  N'  =  ar^-i  +  br^—^  +  cr*-^  + +  ^r  +  q. 

—  =  ar»— 2  +  6r"— 3  +  cr^—^  + +«?  +  —. 

r  r 

Therefore,  the  remainder,  after  dividing  N'  by  r,  is  the  next  digit. 


4:6Q  ADVANCED  ALGEBRA. 

Suppose  iV"  =  ar"-2  +  6r"-'  +  cr*-^  + +  ^, 

N"  n 

=  ar»-3  +  6r«— *  +  cr«-6  + +  ^. 

r  r 

Therefore,  the  remainder,  after  dividing  N"  by  r,  is  the  next  digit. 

Etc.,  etc.,  etc. 

Therefore, 

Hvle, — Divide  the  numher  ly  the  radix,  then  the  quo- 
tient hy  the  radix,  and  so  on  until  the  quotient  becomes  less 
than  the  radix  ;  the  successive  remainders  will  he  the  digits 
of  the  number,  beginning  with  the  units. 

Illustrative  Examples. — 1.  Express  35432  (denary  scale) 
in  the  senary  scale  ;  also,  35432  (senary  scale)  in  the  octary 
scale. 

(1)       6)35432  (2)       8)35432 

6)5905-2  8)2545-4 

6)  984-1  8)212-1 

6)  164-0  8)  14-0 

6)_2_7-2  1-2 

4-3  .-.  35432,.  =  6  =  12014r=8 


.-.  35432r  =  io  =  432012^  =  6 
Explanation  of  (2) : 

35  -T-  8  =  (3  X  6  +  5)  -5-  8  =  23 


8  =  2,  and  7  over ; 


74-^8  =  (7x  6 +  4) -^8  =  46^8  =  5,  and  6  over; 
63  -f-  8  =  (6  X  6  +  3)  -^  8  =  39  -^  8  =  4,  and  7  over; 
72-f"8  =  (7x6  +  2)-s-8=44-^8  =  5,  and  4  over. 

2.  Express  35439  (denary  scale)  in  duodenary  scale  ; 
also,  34439  (nonary  scale)  in  un denary  scale. 

Kote-^The  undenary  scale  needs  a  character  to  represent  ten,  and 
the  duodenary  scale  two  characters  to  represent  ten  and  eleven.  We 
will  represent  ten  by  t  and  eleven  by  e. 

(1)      12)34439  (2)      11)35439 

12 )  2 8 6  9  -  e  11)  2852-5 

12)239-- 1  11)236-8 

12}J^-e  11)18-8 

1-7  1-6 

.-.  34439r  =  io  =  17eler=i2  .-.  35439^  =  9  =  16885^  =  11 


SYSTEMS  OF  NOTATION.  467 

EXERCISE     116. 

1.  Find  the  sum  in  senary  scale  of  4532^  =  6, 

3452r  =  6,  5423^  =  6,  and  3251^  =  6 

2.  Find  the  difference  (octary  scale)  of  3574^  =  8  and 

2756r  =  8 

3.  Multiply  36425r  =  7  by  8;  also  26436,.  =  8  by  10 

4.  Divide  4765^54^  =  11  by  9;  also  2e58if3r  =  i2  by  11 

5.  Express  43250^  =  5  in  the  denary  scale. 

6.  Express  38472r  =  9  in  the  septenary  scale. 

7.  Express  35243^  =  e  in  the  duodenary  scale. 

8.  Express  Set 950r  =  12  in  the  quaternary  scale. 

9.  Find  the  sum  (denary  scale)  of  3472^  =  s  and  5842^  =  10 

10.  Find  the  difference  (nonary  scale)  of  5  ^34^  =  11  and 

6432r  =  7 

11.  What  is  the  radix  of  the  scale  in  which  476^  =  10 

=  2112? 
Suggestion.— Let  r  =  the  radix ;  then  will  2r'  +  r^  +  r  +  2  =  476. 

12.  In  what  scale  is  3  times  134  =  450? 

13.  In  what  scale  is  135^  =  6  =  43  ? 

14.  What  is  the  H.  C.  D.  of  36^  =  8,  48^  =  8,  and  60r  =  8  ? 

15.  Multiply  28r  =  9  by  45r  =  9;  also  square  25^  =  6 

16.  In  what  scale  is  1552  the  square  of  34  ? 

17.  Show  that  35,  44,  and  53  are  in  arithmetical  progres- 

sion in  any  scale  of  notation. 

18.  Show  that  1331  is  a  perfect  cube  in  any  system  of  no- 

tation. 

19.  Show  that  14641  is  a  perfect  fourth  power  in  any  sys- 

tem of  notation. 

20.  Show  that  11,  220,  and  4400  are  in  geometrical  pro- 

gression in  any  system  of  notation. 


468  ADVANCED  ALGEBRA. 

Divisibility  of  Numbers  and  their  Digits. 

866.  Theorem  I. — If  a  number ^  Ny  he  divided  hy  any 
factor  of  r,  r^,  r^  etc.,  respectively  {r  being  the  radix),  it 
will  leave  the  same  remainder  as  when  the  number  ex- 
pressed by  the  last  term,  the  last  two  terms,  the  last  three 
terms,  etc.,  is  divided  by  the  same  factor. 

Dem(mstration :  Suppose  x  a  factor  oi  r,  y  a  factor  of  r^,  and  z  a 
factor  of  r^,  etc. 

iV=  ar«-*  +  Jr~-2  + +  pr^  +  qr  +  s. 

Now,  X  is  certainly  a  factor  of  every  term  of  N,  except  s;  y,  a, 
factor  of  every  term,  except  qr  +  s;  and  z,  a  factor  of  every  term, 
except  pr^  +  qr  +  s,  etc.    Therefore, 

N  s 

1.  —  =  an  integer  h — . 

rt   iV  .  ,  qr  +  8 

2.  —  =  an  integer  + . 

N^  t)r^  +  <7  7*  4-  s 

3.  —  =  an  integer  + ;  which  was  to  be  proved. 

z  z 

867.  Car, — In  the  decimal  system  of  notation, 

1.  A  number  is  divisible  by  any  factor  of  10,  if  the 
units^  digit  is  divisible  by  that  factor, 

2.  A  number  is  divisible  by  any  factor  of  100,  if  the 
number  expressed  by  the  last  two  figures  is  divisible  by 
that  factor. 

S.  A  number  is  divisible  by  any  factor  of  1000,  if  the 
number  expressed  by  the  last  three  figures  is  divisible  by 
that  factor. 

868.  Theorem  II. — The  difference  between  a  number 
and  the  sum  of  its  digits  is  divisible  by  the  radix  less  one. 

Bemonstration : 

Let  N  =z  ar"— >  +  &r»-2  + +  pr^  -^  qr  ■{•  s  =  any  number; 

then,  a  +  6+....4-jp  +  g'  +  5  =  the  sum  of  the  digits. 

Now,  a  (r«-»  —1)  +  b  (r»-9  -1)  + +  p  (r^— 1)  +  q  (r— 1)  =  the 

difference  between  the  number  and  the  sum  of  its  digits,  and  every 
term  is  divisible  by  r  —  1. 


DIVISIBILITY  OF  NUMBERS,  469 

869.  Cor. — In  the  decimal  system  of  notation, 

The  difference  letween  a  number  and  the  sum  of  its 
digits  is  divisible  by  9  or  S. 

870.  Theorem  III, — A  number,  N,  divided  by  r—1, 

leaves  the  same  remainder  as  the  sum  of  its  digits  divided 

by  r  —  1,  r  being  the  radix. 

Demonstration. — Put  s  for  the  sum  of  the  digits;  q  and  q'  for 
the  quotients;  and  c  and  c'  for  the  remainders. 

1.  N=  q{r-l)  +  c 

2.  s  =  q'  (r  —  l)  +  c' 

.*.    N—s  =  (q  —  q')  (r  —  1)  +  (c  —  c'). 
Now,  iV  —  s  is  divisible  by  r  —  1  [T.  II],  and  (q  —  q') (r—1)  is 
evidently  divisible  by  r  —  1;  therefore,  c  —  c'  is  divisible  by  r  —  1. 
But  c  and  c'  are  each  less  than  r  —  1 ;  hence,  c  —  c'  =  0,  or  c  =  c', 

871.  Car, — In  the  decimal  system, 

A  number  is  divisible  by  9,  if  the  sum  of  its  digits  is 
divisible  by  9. 

872.  Theorem  IV, — If  from  a  number,  N,  we  sub- 
tract the  digits  of  the  even  powers  of  r,  and  add  those  of 
the  odd  powers,  the  result  will  be  divisible  by  r  +  1. 

Demonstration. — Let  iV=  ar*  +  br^  +  cr^  +  dr  +  e. 
Add      —a     +  b     —  c     +  d   —  e^ 
then,  a(r*  —  1)  +  b (r^  +  1)  +  c{r^  —  1)  +  d{r  +  1),  the  result,  is  divis- 
ible by  r  +  1,  since  every  term  is  divisible  by  r  +  1. 

873.  Theorem,  V, — If  a  number,  N,  be  divided  by 
r  + 1,  the  remainder  will  be  the  same  as  when  the  differ- 
ence between  the  sums  of  the  digits  of  the  even  and  odd 
powers  of  r  is  divided  by  r-\-l. 

Demonstration.— Put  d  for  the  difference  between  the  sums  of  the 
digits  of  the  even  and  odd  powers  of  r ;  g  and  q'  for  the  quotients ; 
and  c  and  c'  for  the  remainders ;  then  will 

iV=  g(r  +  1)  +  c, 
and  d  =  q*  {r  +  \)  +  c'. 

.-.    N-d  =  {q-q'){r  +  l)  +  c-c'. 
Now,  N  —  d  is,  divisible  by  r  +  1  [T.  IV],  and  {q  —  q')  (r  +  1)  is 
evidently  divisible  by  r  +  1.    Therefore,  c  —  c'  is  divisible  by  r  +  1. 
But  c  and  c'  are  each  less  than  r  +  1 ;  hence,  c  —  c'  =  0,  or  c  =  c'. 


470  ADVANCED  ALGEBRA. 

874.  Cor, — In  the  decimal  system  of  notation, 
A  number  is  divisible  by  11,  if  the  difference  between 
the  sums  of  the  digits  in  the  even  and  odd  places  is  divis- 
ible by  11, 


Even  and  Odd  Numbers. 

875.  An  even  number  is  a  number  that  is  exactly  di- 
yisible  by  2. 

876.  An  odd  number  is  a  number  that  is  not  exactly 
divisible  by  2. 

877.  If  we  let  x  represent  any  integral  number  in- 
cluding zero,  and  regard  zero  as  an  even  number,  it  be- 
comes evident  that  the  general  formula  for  an  even  num 
ber  is  2  x,  and  for  an  odd  number  2  ic  + 1. 

878.  Theorem,  I, — The  sum,  of  any  number  of  even 

numbers  is  even. 

Demonstration. — Let  2  a:i ,   2  iCa ,  2  X8 , 2xn   represent  n  even 

numbers ;  then  will  their  sum  be 

2a;i  +  2a:a  +  2a:8  + +  2xn  =  ^{xi  +  x^ -^  Xt  + +  a^n) 

an  even  number. 

879.  Jlieorem  II. — The  sum  of  an  even  number  of 

odd  numbers  is  even. 

Demonstration.— Let  2a;i  +  1,  2a;a  +  1,  2a;8  +  1  + +  2a;an  +  1 

represent  2»  odd  numbers ;  then  will  their  sum  be 

(2a;x  +  1)  +  {2xu  +  1)  +  (2xs  +  1) +....  + (2x2n  +  t)  = 

2xi  +2Xi  +  20^8  + +  2a;an+  2n  = 

2  (a^i  +  a:a  +  iCa  + . . . .  +  x^n  +  n),  an  even  number. 

880.  Theorem  III, — The  sum  of  an  odd  number  of 
odd  numbers  is  odd. 

Demonstration. — Let   2xi  +  1,   2ira  +  1,  2a;8  +  1, 2a;9«+i  +  1 

represent  271  +  1  odd  numbers ;  then  will  their  sum  be 

(2a;i  +  1)  +  (2a;a  +  1)  +  (2a:8  +  1)  +  . . . .  +  2a;3«+i  +  1)  = 

2xi  +  2a;a  +  2a;8  + +  Sa^sn+i  +  27i  +  1  = 

2{xi-\-Xi  +  Xf\- +  x%n+\  +  n)  +  1,  an  odd  number. 


EVEN  AND  ODD  NUMBERS.  471 

881.  Theorem  IV, — The  sum  of  an  equal  even  number 
of  even  and  odd  numbers  is  even. 

Demonstration.— Let  (2xi  +  1)  +  (2a;a  +  1)  + . . . .  +  (3a:an  +  1)  = 
the  sum  ot  %n  odd  numbers ; 

and  2a;'i  +  2a;'a  + +2ic'a«         = 

the  sum  of  2  n  even  numbers ;  then  will  their  sum  be 

{2(a;i  +  a:'x)  +  1}  +  1 2  (a^a  +  ic'a)  +  If  + +  \'i{Xin  + x'^n)  +  1}, 

which  is  even  [T.  II]. 

882.  Thefyrem  V. — The  sum  of  an  equal  odd  number 
of  even  and  odd  numbers  is  odd. 

Demonstration. — Let  (2a;i  +  1)  +  (2a;a  +  1)  + +  (2a'9»  +  i  +1)  = 

the  sum  of  2  ?i  +  1  odd  numbers ; 

and  2a;'i  +  2a;'a  + +  2a;'2»  +  i         = 

the  sum  of  2n  +  1  even  numbers ;  then  will  their  sum  be 

{2{Xi  +  x\)  +  1\  +  {2(2:2 +  a;'a)  +  1}  + +  {2{Xin+i  +  x'^n  +  i)  +1}, 

which  is  odd  [T.  IIIj. 

883.  Theorem  VI. — The  difference  bettoeen  two  num- 
bers, if  both  are  odd  or  both  even,  is  even. 

Demonstration. — 1.  Let  2  x  and  2  x'  be  two  even  numbers. 
Their  difference  is  2  a;  —  2  a;'  =  2  (a;  —  a;'),  which  is  even. 
2.  Let  2  a;  +  1  and  2  a;'  +  1  be  two  odd  numbers. 
Their  difference  is  (2  a;  +  1)  —  (2  a;'  +  1)  =  2  a;  —  2  a;'  =  2  (a;  —  x% 
which  is  even. 

884.  Theorem;  VII, — The  difference  between  an  odd 

and  an  even  number  is  odd. 

Demonstration. — Let  2  a;  +  1   be  any  odd  number,  and  2  x'  any 
even  number. 

Their  difference  is  (2  a;  +  1)  —  2aJ'  =  2  (a;  —  a;')  +  1,  which  is  odd. 

885.  Theorem  VIII,— The  product  of  any  number  of 

even  numbers  is  even. 

Demonstration. — Let  2  a:, ,  2  a^a ,  2  a^s ,    . . .  2  a:„  be  n  even  numbers. 
Their  product  is  2(2*-ia;i ,  Xi,  Xz, a;„),  which  is  even. 

Cot, — Any  power  of  an  even  number  is  even, 

886.  Theorem  IX, — The  product  of  any  number  of 
odd  numbers  is  odd. 


472  ADVANCED  ALGEBRA. 

Demonstration. — Let  SiCi  +  1,  2a;a  +  1, 2a;n  +  1  be  n  odd  num- 
bers. It  is  evident,  from  the  nature  of  multiplication,  that  the  prod- 
uct of  these  numbers  will  contain  the  factor  2  in  every  term,  except 
the  last,  which  will  be  1.  That  is,  the  product  will  have  the  form  of 
2  a;'  +  1,  which  is  odd. 

887.  Cm; — Any  power  of  an  odd  number  is  odd. 

888.  Theorem  X, — The  product  of  any  number  of  odd 
and  even  numbers  is  even. 

Demonstration. — The  product  of  the  odd  numbers  is  odd  [T.  IX], 
and  may  be  represented  by  2  a;  +  1. 

The  product  of  the  even  numbers  is  even  [T.  VIII],  and  may  be 
represented  by  2  x'. 

.-.  The  entire  product  is  2x' {2x+l)  =  2(xx'+x'),  which  is  even. 

Example. — It  is  required  to  divide  one  dollar  among  15 
boys,  giving  to  each  boy  an  odd  number  of  cents.  Is  this 
question  possible  ? 


Prime,  Composite,  Square,  and   Cubic  Numbers. 

I.   Definitions. 

889.  A  Prime  Number  is  a  number  that  can  not  be 
produced  by  multiplying  together  factors  other  than  itself 
and  unity. 

A  prime  number  is  divisible  only  by  itself  and  unity. 

890.  A  Composite  Number  is  a  number  that  may  be 
produced  by  multiplying  together  other  factors  than  itself 
and  unity. 

A  composite  number  is  divisible  by  other  factors  than 
itself  and  unity. 

891.  A  Square  Number  is  one  that  may  be  resolved 
into  two  equal  factors. 

892.  A  Cubic  Number  is  one  that  may  be  resolved  into 
three  equal  factors. 


PRIMES.  473 

893.  Two  or  more  numbers  are  prime  to  each  other 
when  they  have  no  common  factor,  except  unity. 

2.   Primes. 

894.  Theorem  I, — The  number  of  primes  is  unlimited. 
For,  let  n  be  the  number  of  primes,  and,  if  n  is  not 

unlimited,  let  p  be  the  greatest  prime  number.  Then  will 
2x3x5x7xllX  ....  XphQ  divisible  by  all  primes  not 

greater  than  jo  ;  and  (2  X  3  X  5  X  7  X  11  X X  i?)  -f  1 

not  be  divisible  by  any  prime  not  greater  than  j9.  There- 
fore, (2X3X5X7X11X Xi?)  +  1  is  itself  a  prime 

greater  than  p,  or  is  divisible  by  a  prime  greater  than  p. 
In  either  case,  p  is  not  the  greatest  prime.  Therefore,  n 
is  unlimited.  

895.  Theorem  II, — Every  prime  number,  except  2  and 
3,  belongs  to  the  form  6  ;^  ±  1. 

For,  every  number  evidently  belongs  to  one  of  the 
forms  Qn,  6  w  +  1,  6  w  +  2,  6  ?^  +  3,  6  tj  +  4,  or  6  w  +  5, 
in  which  n  may  be  any  integer  including  0.  Now,  6w, 
6  ?i  +  2,  and  6  w  +  4,  are  each  divisible  by  2,  and  6  tj  +  3 
by  3  ;  hence,  these  forms  are  composite,  except  when  ^  =  0 
in  6  ?^  +  2  and  6^  +  3,  in  which  case  we  have  the  primes 
2  and  3. 

The  only  forms  remaining  to  contain  primes  are  Qn-\-l 
and  6^  +  5.  But  6  w  + 5  =  (67^  + 6)  - 1  =  6  (?i  +  1)-1 
=  6  7^'  ■—  1.  Therefore,  the  general  form  6  7^  ±  1  contains 
all  primes,  except  2  and  3. 

Scholium, — It  must  not  be  inferred  from  this  propo- 
sition  that  all  numbers  expressed  by  6  7i  ±  1  are  prime. 
Thus,  when  w  =  4,  6^  +  1  =  25;  and  when  n  =  11, 
6  71  —  1  =  65. 


474  ADVANCED  ALGEBRA. 

Cor, — Every  prime  above  S,  increased  or  diminished 
hy  unity,  is  divisible  by  6, 


896.  Theorem  III, — iVo  rational  formula  can  repre- 
sent primes  only. 

For,   if    possible,    let    a-{-bx-\-cc(^-}-dcc^-\- be 

prime  for  all  values  of  x. 

When  X  =  m,  let  a-\-bx-^cx^-\-da?-\- =i?; 

then,         p  =  a-\-bm-\-cm^-{-dm^-\- 

When  X  =  m-{-np,  leta-\-bx-\-cx^-\-da^-\- =g ; 

then,         q  =  a-{-b{m-{-  np)  -{-  c{m-\-  npf  + 

d(m-\-  npY  +  . . . . 

=  a-\-bm-\-cm^-\-dm^-}- -\-rp 

=z p  -{-  r p  =  p  (1  -\-  r)f  a  composite  number. 

897.  Scholium, — The  form  n^-\-n-\-4:l  is  prime  for  all 
values  of  n  from  0  to  39  inclusive,  and  the  form  2  ^^  +  29 
for  all  values  of  n  from  0  to  28  inclusive.  These  forms 
have  been  discovered  by  trial,  and  are  not  demonstrable. 


898.  Theorem,  IV, — If  a  number  is  not  divisible  by  a 
factor  equal  to  or  less  than  its  square  root,  it  is  a  prime. 

For,  let  N=xxy  be  any  number  not  prime.  Then, 
if  x  =  y,  W=  /,  and  y  =  VW,  But,  if  ic  >  VW,  then 
y  <  ViV,  since  x  X  y  =  J^-  But  JV  is  divisible  by  y. 
Therefore,  if  iV  is  not  prime,  it  is  divisible  by  a  factor 
equal  to  or  less  than  VJSf,  Hence,  too,  if  a  number  is  not 
divisible  by  a  factor  equal  to  or  less  than  its  square  root, 
it  is  prime. 

3.  Composites. 

899.  Theorem  I. — If  a  number  is  a  factor  of  the  prod- 
uct of  two  numbers  and  is  not  a  factor  of  one  of  them,  it 
is  a  factor  of  the  other. 


COMPOSITES.  475 

Thus,  let  a;  be  a  factor  of  a  h,  and  not  a  factor  of  a ; 
then  will  it  be  a  factor  of  5. 

For,   —  may  be  reduced  to  a  terminating  continued 

fraction  [854].     Let  —  be  the  conyergent  next  in  yalue 

to  -.     Then,  ^  ^  —  =  —  [848,  P.] ;  whence, 
X  q         X        qx  ^  -^^ 

px'^aq  =  l;  and  hpx  ^  abq  =:  h. 

Now,  hp X  and  abq  ?iie  each  divisible  by  x ;  therefore, 
their  difference,  h,  is  divisible  by  x, 

900.  Car, — If  a  number  is  prime  to  each  of  two  or  more 
other  numbers,  it  is  prime  to  their  product. 


901.  Theorem  II. — Every  composite  number  may  be 
resolved  into  one  set  of  prime  factors  and  into  only  one 
set. 

1.  Any  composite  number  {N)  is  the  product  of  two 
or  more  factors  each  less  than  N,  which  are  all  composite, 
all  prime,  or  some  composite  and  some  prime.  As  many 
of  these  as  are  composite  are  again  resolvable  into  other 
factors  less  than  themselves,  and  so  on,  until  no  factor  is 
further  resolvable  into  factors  less  than  itself  and  greater 
than  unity,  at  which  stage  all  the  factors  are  prime. 

2.  Let  one  set  of  prime  factors  oi  N  he  a,  b,  c, , 

and,  if  possible,  let  another  set  be  ^,  g',  r, ;  then  will 

axbxcx =pXqXrX 

Now,  suppose  a  different  from  q,  r, ,  then  it  is  not 

contained  in  q  X  r  X [900]  ;  it  must,  therefore,  be 

contained  in  p,  but  this  can  only  be  when  a=p,  since  p 

is  a  prime.     But,  it  a=p,  bXcX =qXrX ; 

from  which  it  follows  as  before  that  b  is  identical  with  one 
of  the  factors  in  qXr  X ;  etc. 


476  ADVANCED  ALGEBRA. 

902.  Theorem  III, — The  product  of  any  r  consecutive 

numbers  is  divisible  hy  [r. 

^       n(n  —  l)(n  —  2) (n  —  r-\-l)   .     , ,  ,     , 

For,  — ^ — 1 ^^ ' — -  IS  the  product 

of  r  consecutive  numbers  divided  by  [r,  and  it  is  also  the 
number  of  combinations  of  n  things  taken  r  together, 
which  is  evidently  a  whole  number. 

903.  Cor,  J.— The  coefficient  of  the  {r  +  l)th  term  of 

^,     ,  .        ....  .    n{n-l)(n-2)....{n-r-\-l) 

the  bmomial  theorem  is  -^ — , 

[595] ;  therefore, 

The  coefficient  of  every  term  of  the  binomial  theorem  is 

integral  when  n  is  a  positive  integer, 

904.  Cor,  2, — If  we  represent 
n(n-X)(n-2)         (n-r±V)  ^^  ^^  .^  ^^jj^^^  ^^^^^ 

All  factors  of  the  numerator  that  are  prime  and  are 
greater  than  r  are  divisors  of  q. 


905.  Theorem  IV, — Fermafs  Theorem.  If  p  be  any 
prime  number,  and  a  be  a  number  prime  to  p,  then 
a^~^  —  1  will  be  divisible  by  p. 

Demonstration  :  a^  =  [1  +  (a  —  1)]p 

=^  1  +  p{a-l)  +  ^^^~^\a-lf  +  . . . .+  {a-  V)p         (A) 

.-.    aP  —  {a  -  \)P  -1  =  p{a  -1)  +  --^g        +  ^*^^- 

=  a  multiple  of  p  [901.    140,  P.].      "~  (B) 

Let  a  =  3,  then 

ap  ~.{a  —  l)P  —  \  =  2/'  —  2  =  a  multiple  of  p. 
Let  a  =  3,  then 

aP  -  (a-\)P  -1  =  ^p  -2p  -I  =  {^P  -  3)  -  {2P  -  2) 
=  a  multiple  of  p. 
...    Si*  —  3  is  a  multiple  of  p  [157,  P.]. 


PERFECT  SQUARES,  477 

By  continuing  this  process,  it  may  be  shown  by  induction  that 
aP  —  a  is  a  multiple  of  p. 

But  aP  —  a  =  a(aP—^ —  1)  and  a  is  prime  to  p;  therefore, 
aP-^  —  1  is  divisible  by  p. 


Perfect  Squares. 

906.  Theorem  I. — Uvery  square  number  is  of  the  form 
3  m  or  3  m  +  1. 

For,  every  number  is  of  the  form  of  3x  or  3  a:  ±  1. 

Now,  {3xY  =  dx^  =  d{3x^)=z3m;  and 
{3x±lY=(9af±6x  +  l)  =  3{3x±2)-\-l  =  3m-{-l. 


907.  Theorem  II. — Every  square  number  is  of  the 
form  4^m  or  4  m  +  !• 

For,  every  number  is  of   the  form  of    ^x,    4cX-\-l, 
4a:  +  2,  or  4a-  +  3. 

Now,  {4.xf  =  16a^  =  4.(4:3^)  =  4:m; 
{4:X  +  iy  =16a^-^    82;  +  l  =  4(4a;2  +  2:r)  +  l 

=  4m  +  l; 
{4:X  +  2Y  =  16a^-\-lGx  +  4.  =  4:  (4:  a^  +  4:  x -\- 1) 

=  4:m; 
and    {4:X-{-3y  =  16 a^ -\- 24: x -{- 9  =  4: {4: x^ -{- 6 x -\- 2)  + 1 
=4/7i  +  l. 

908.  Theorem,  III, — Uvery  square  number  is  of  the 
form  5  m  or  5  m  ±  1. 

For,   every  number  is  of   the  form  6x,    5  ic  ±  1,  or 

5  a:  ±  2. 

Now,  (5 xY  =z  25x^  =  5  {6a^)  =  5m; 
(5a;  ±1)2  =  25a;2±10a;  +  l  =  5(5a;2±2^)  +  l 

=  5m-\-l ; 
and    (5.'c±2)2  =  25a;2±20a;  +  4  =  5  (5  3^  ±  4:X  +  1) -1 

=  5  m  — 1. 


478  ADVANCED  ALGEBRA, 

909.  Theorem  IV, — If  a^-\-l^  =  (^  when  a,  h,  and  c 
are  integers,  then  luill  ale  he  a  multiple  of  60, 

For,  1.  a^  and  h^  can  not  both  be  of  the  form  3  m  +  1, 
else  would  c^  be  of  the  form  3m +  2,  which  is  not  a 
square.     Therefore,  either  a  or  J  is  a  multiple  of  3  [906]. 

2.  a^  and  h^  can  not  both  be  of  the  form  of  4^  +  1, 
else  would  c^  be  of  the  form  4m +  2,  which  is  not  a 
square.  Therefore,  either  a  ov  h  must  be  a  multiple  of  4, 
or  each  of  them  a  multiple  of  2  [907].  In  either  case, 
ahc  is  a  multiple  of  4. 

3.  a^  and  ¥  can  not  both  be  of  the  form  5  m  + 1  or 

5  m  —  1,  else  would  (^  be  of  the  form  5  m  ±  2,  which  is 
not  a  square.  Therefore,  either  a^  or  h^  must  be  of  the 
form  5  m,  or  one  of  the  form  bm-\-l  and  the  other  of 

6  m  —  1  [908].  In  the  former  case,  either  a  or  5  is  a 
multiple  of  5,  and  in  the  latter,  c  is  a  multiple  of  5,  and 
in  either  case,  ahc  is  a  multiple  of  5. 

4.  Since  ale  is  a  multiple  of  3,  4,  and  5,  and  these 
numbers  are  prime  to  each  other,  ale  is  a  multiple  of  60. 

Scholium, — By  means  of  this  theorem  and  the  formula 
a  =  V{c  -{-l){c  —  I),  rational  values  of  a,  I,  and  c  may 
he  determined  ly  inspection  that  will  satisfy  the  equation 
a^^¥=:(^. 

910.  Problem.     To  determine  the  rational  value  of  x 

that  will  render  x^  +px  +  q  a  perfect  square. 

Solution :  Let  x^  +px  +  q  =  (x  +  mf 
then,  x^  +  px  +  q  =  x^  +  ^mx  +  m^ 

whence,  x  = ^ ,  in  which  m  may  have  any 

rational  value  from  --  oo  to  +  oo . 

Illustration. — What  value  of  x  will  render  a;^  —  7  a;  +  2 
a  perfect  square  ? 


PERFECT  SQUARES.  479 


Solution :  Here  p  =  —  7,  g  =  2,  and  let  m  =  5, 

25-2  23  ,6 

then,  X  =  _^_-^Q  =  1117  =  -  1 17 

.      r,        o       529       161      „      3844       /62\2 

911.  Cor,  1, — For  m  >  Vq  and  2m  <p,  or  m  <  Vq 

and  2m  > p,  m  being  positive,  x  will  be  positive, 

912.  Cor.  2. — Put  m^  —  q  =  n{p  —  2m);  then 

q  =  m^  —  n  {p  —  2  m) ; 
X  =  n,  an  integer ;  and 
oi?-\-px-\-q  —  x^  -\-  p  X  -\-  m^  —  n  (p  —  2  m) 
=  {n-\-  my,  an  integer. 

7f?  —  0 

913.  Cor.  3.— Put  X  = -^  —  —  m;  or 

p  —  2m 


m=^  ±  a/^  -  q,  then, 


x^-\-px-\-q  =  0,  and  a;  =  —  |  :f  a/ ^ 
which  conforms  to  Art  SSI. 


Perfect  Cubes. 


914.  Theorem  I. — Every  cube  is  of  the  form  4om  or 
4m  ±1. 

For,  every  number  is  of  the  form  4:X,  4:X-{-l,  4  a;  +  2, 
or  4  a;  +  3. 

Now,  {4:xY  =  64:X^  =  4.(16x^)  =  4:m; 
{4:X-\-iy  =  64:a^-\-A83^-\-12x-{-l 

=  4  (16  a:^  +  12  rc2  +  3  a;)  +  1  =  4  m  +  1 
(4a;4-2)^  =  64a:3_^96^2_j_43^_|_8 

=  4(16x3  +  24a;2  +  12a;  +  2)=4m 
(4a;  +  3)^  =  64ar^  +  144a;2  +  108a;  +  27 

=  4:{Ux^-\-36x^-i-27x+'7)-l  =  4:m-l 


480  ADVANCED  ALGEBRA. 

915.    Problem.    To  determine  rational  values  of  x  that 
will  render  a?  +  px^  ■\-  qx  +  r  a  perfect  cube. 

Solution :  Put  x^  -{■  px"^  +  qx  ^^  r  =  {x  +  mf  \  then, 

(^  — 3m)a;2 +  (5'-3m2)a;  + (r  — m3)  =  0.  (A) 

1.  Put       bm—p,   or  m  =  -^p\  then 

_    m^  —  r   _  jp«-27r 
^-  q-'dm^-  27q-9p' '  ^^"^ 

(pS 27 r        «  \' 

~  V       27g-9i?2       )  '    ■ 
Cor.—If9pq-27r-'2p^  =  0,   or    r  =  ^^^Il^^\ 

3  Q  '^* 

x^-\-p^  +  qx-\-r  =  0,  and  x  =  ^^—^^. 

2.  Put  m*  =  r,   and  suppose  r  =  ri%  then  m  =  ri ;   and  (A)  will 
become   (i?  —  3 ri) x^  +  {q  —  B Ti^) x  =  0;  whence, 

x=      ^o      ;  and  x^+px^  +  qx  +  r  =  x^+px^  +  qx  +  ri^; 


and  (re  +  m)^  =  (^^3^^)  .    Therefore, 


^  — 3ri 


_o        ) 

Cor, — If  q  =p n,  x=  —ri,  and 

7? -\-p :i^ -\-qx-\-rx  ^^' 


,  Scholium, — Other  values  under  particular  suppositions 
may  he  obtained  hy  putting  3m^  =  q  =  3qi^. 

EXERCISE     117. 

1.  Find  which  of  the  following  numbers  are  prime  : 
19r,  251,  313,  281,  461,  829,  957. 

2.  Find  the  least  multiplier  that  will  render  3174  a 
perfect  square. 

3.  Find  the  least  multiplier  that  will  render  13168  a 
perfect  cube. 


PERFECT  CUBES.  481 

4.  Find  which  of  the  following  numbers  are  divisible 
by  9,  which  by  11,  and  which  by  both  9  and  11 :  11205, 
24530,  342738,  25916,  558657. 

5.  Show  that,  if  ^  +  3'  is  an  even  number,  then  is  j9  —  g 
also  an  even  number,  provided  p  and  q  are  integral. 

6.  Show  that  every  cube  number  is  of  the  form  7  w  or 
7/^±l. 


7.  Find  such  a  value  of  x  as  will  render  the  wax 
rational. 

Suggestion. — Put  ^/a  x  =  p. 


8.  Find  such  values  of   a;   as  will   render    vax-\-h 
rational. 

9.  Prove  that  2*"  —  1  is  a  multiple  of  15. 

'  10.  Show  that  no  square  number  is  of  the  form  3  »  —  1. 

11.  Show  that  n{n-\-l){^n-\-l)  is  divisible  by  6. 

12.  Show  that  {n^  -f  3)  (w*+  7)  is  divisible  by  32,  when 
n  is  odd. 

13.  Show  that  n^  —  n  is  a  multiple  of  30. 

14.  Show  that  the  fourth  power  of  any  number  is  of 
the  form  5  m  or  5  m  +  1. 

15.  Every  even  power  of  every  odd  number  is  of  the 
form  8  w  + 1' 

16.  Show  that  every  square  can  be  expressed  as  the 
difference  between  two  squares. 

17.  Show  that  a'  -\-a  and  a'  ^a  are  even  numbers. 

18.  Show  that  every  number  and  its  cube  leave  the 
same  remainder  when  divided  by  6. 

19.  If  ?^  >  2,  show  that  n^  —  5n^-{-4:n  is  divisible  by 
120. 

20.  It  n  is  a,  prime  number  greater  than  3,  show  that 
n^  —  1  is  divisible  by  24. 


482  ADVANCED  ALGEBRA. 

21.  Find  such  a  value  of  x  as  will  render  Vox  rational. 

Suggestion. — 

Let  ^/a  x  =  p,  any  rational  quantity,  then  will  re  =  —  . 


22.  Find  such  a  value  of  a;  as  will  rationalize  vax-{-h. 
Suggestion.— Put  ^/ax  +  b  =  p,  and  prove  x  =  — . 

23.  Find  such  a  value  of  x  as  will  rationalize 

Suggestion. — Put  \/ax^  +  bx  =  px,  and  prove  x  =     g_    . 

24.  Find  such  a  value  of  x  as  will  rationalize 

Suggestion. — 

Put  '\/ax^  +  bx  +  c^  =  px  +  c,  and  prove  x  =    ^_    ^  . 

25.  Find  such  a  value  of  a;  as  will  rationalize 

Va^a^-\-bX'^c. 
Suggestion. — 

Put  ^/a^ x^  +  bx  +  c  =  ax  +  p,  and  prove  a;  =  ,  _  ^ — . 


26.  Find  such  a  value  of  a;  as  will  render  Vax^-{-bx-\-c 
rational  when  J^  —  4  «  c  is  a  perfect  square. 

Suggestion.— Put  \/ax^  +  bx  +  c  =  0,  and  b^  —  4ac=z  q^,  and 

-b  ±q 
prove  a;  =  -2^. 

27.  Find  such  a  value  of  x  as  will  rationalize 


, ,  p^-b 

Suggestion.— Put  ^a^  ->cbx^  =  px,  and  prove  x  =      ^     . 

28.  Find  such  a  value  of  a:  as  will  rationalize 

Vaa^-^ba^-\-cx-\-dK 
Suggestion.-  ^,_^^^ 


Put  Va^  +  6a;«  +  ca;  +  d«  =  2^  a:+d,  and  prove  x  =  -^ 


d« 


ANSWEES. 


1.  l+x+x'+x' 


Exercise 

87. 

2. 

1 
3  + 

2          4 

'^'U 

4.  1+x- 

-x^- 

-7? 

6. 

-d  +  5x- 

-2x^- 

-Saf 

81 


3.  1+x-x^-x* 
5.  l-x-x^  +  2a^ 

a       a'       a* 

9.  l+a:  +  a;2— a;8 


Exercise  88. 

1-  2-  ^^+  4^^+  3^6^'  2.  1+  i^-  |^«-  Ix^ 

^    ,      1         109   ,      109    3  ^    .       1  1    5      5    , 

3'  ^+  6  ^-  216^  +  3888^  ^'  ^+  3  ^-  9  ^  "^  81^ 


27         3.27«     "3.27* 

^   a      1  23  25  „     I  .      X  x^     ^      x^ 

^'^^r2^^27T2^-2A2^  '-"^^^-^^I^l 

_ia;  a;'  7  ^  a:*ic«5a;' 

3a§       9a^       81al  ^a^       ga*^       81a8 

Exercise  89. 

1.  Divergent.       2.  Convergent;  divergent;  divergent. 
3.  Convergent.      4.  Convergent.      5.  Divergent. 

6.  Convergent;  divergent;  divergent. 

7.  Divergent;  convergent;  divergent.  8.  Convergent. 

Exercise  90. 

1.  x^  +  Qx^  +  llx  +  Q  2.  a;3_2a;2-9a;  +  18 

3.  a;4+2a;3-7a;«-8a;  +  12  4.  a:4-37a;2-24a;+180 


484  ADVANCED  ALGEBRA. 

5,  a;4  +  8a:3  +  24a:2  +  32a:  +  16  6.  a;4_20a;3  +  150a:2-500a;+625 

7.  16  a;4-16  2:3-64.1:2 +  4 a; +15         8.  (a: +  2)  (a; +3)  (a: +  4) 

9.  (a:-3)(a:-4)(a:+5)  10.  (a:  +  2)(a:+3)(a:  +  l)(a;-l) 

11.  (a;  +  2)(a:-3)(a:  +  4)(a:-5)  12.  (a;  +  2)(a;-2)(a;+3)(a;-3)(a;+4) 

Exercise  91. 

1.  a4_i2a3a;2  +  54a2a:4_i08aa;«  +  81a:4 

2.  32  +  400  a:  +  2000  x^  +  5000  x^  +  6250  a:*  +  3125  a:* 

3.  a:6-18aa:^  +  135a2a:4_540a3a;3+i2i5a4a:2_i458a6a:+729a« 

4.  128  a:!^  +  2240  x^^  + 16800  a;io  +  70000  a? + 175000  a:«  +  262500  a:^  + 

218750  a:2  + 78125 

5.  a;4-40 xl  +  700 3:^-7000 a;t  +  43750 a:2-175000a:l  +437500 x 

-  625000  a:i  + 390625 

6.  2187  a;¥  +  5103  at  a:^  ^.  5103  ^3  ^^.^  +  2835  ai  a:l  +  945  a«  a:^  + 

189  a¥a:f  + 21  a9a:§+a¥ 

7.  l__a:-ga:2-_a:3 

8.  at  —  -5-  a~i  ^  —  77  ^~  ^  ^^  —  01  ^~~  ^  ^ 

o  9  81 

9.xt-|x-i-|x-5-j|g^-| 

10.  a;— 4  —  —  aa;~4  +  —a^x—i  —  r-^a^a:~4 

-o  o  Id 

11.  a~%x~%  —  —  a~V  Ja:~  (T  +  —  a~V  h^x~^  — 

12.  a:— ^  +  -^  aix—^  +  -^aia;— V  +  — a^a;"^ 

14.  8-06225;  8*94427;   7*006796;  5-000980 

15.  2449440a.-*  16.  -  -^^a^  17.  -  ^a""^^' 

,3,  ^r(r+l)(r+2)(r  +  3)^_,,^,,^_. 

Li 

19.  2^  20.  iJ^  21.  0 

343 


Exercise  92. 

1.  n(n  +  l);  ■|n(n  +  l)(n+2)  2.  (2n-l)«;  ■|n(4w«-l) 


AKSWURS.  485 

3.  ^n(n  +  l)',  ^n(n  +  l)(n  +  2)       4.  w(n  +  3);  ^n(n  +  l)(2n  +  7) 

5.  w2;   ^nin+l){2n+l)  6.  2n{2n-l);  -n{n  +  l){4n-l) 

o  o 

1.  n{n  +  l)in+2);  -^n(n  +  l){n  +  2)(n+S) 

8.  n(n  +  4)(n+S);  -jn(n  +  l){n  +  8)in  +  d) 

9.  n(n  +  2)(n  +  lf;  ^n(n  +  l)(n  +  2)(n  +  d)i2n  +  d) 

10.  {2n  +  l){2n  +  3){2n  +  5);  -i(2w  +  l)(2n  +  3)(2w  +  5)(2w  +  7) -IS^ 
o  o 

n  1  J./        1        1^ 1 1 1_\     11 

3?i  +  l'    3  3  V   "^  2  "^  3       w  +  l       w  +  2      n  +  dj'   18 

13    iri____i___V  1 

8  V4       2(?i  +  l)(n  +  2)y'  32 

^ 6n  +  ll  ^  21 

180       12(2n  +  l)(2n  +  3)(2w  +  5)'   180 

15.  i  _  -_1^+^;  4  16.  ^(9n^  +  10n^-Sn-4) 

4       2(ri  +  l)(w  +  2)'    4  12^  ^ 

^^   n(n  +  l)(n  +  2)(n+3)  ^g    _w_ 

[4  ■  n  +  1 

19.  n  +  1 K;  20.  2-978809 

Exercise  93. 

1.  x  =  y-y^  +  y^-y*+ 2.  x  =  y+ -^y^+ -^y^+  24 2/*+ 

3.  x  =  y—2y^  +  Sy^—4:y^+ 4.  x  =  y—y^+y^—y''  + 

b.x  =  (y-1)  -  1  (i/-l)2  +  1  (^-1)3  _  ^  (y_l)4  + . . . . 

6.x  =  (y-1)  +  2iy-lf  +  liy-lf  +  30(2/-l)*+ .... 

7.  a;  =  i-|2  +  |^-^  =  '17590144  8.  a;  =  -00999999 


Exercise  94. 

1.^  =  3,  ^=-1;  .,    i~^    ,  2.  »  =  2,  q  =  2;  ,     ^^"*"^^   ^ 

^  ^  1— 3a;  +  a;2  ^        '  ^  1— 2aj— 2a;2 

o    *,      ;t  Q         2  +  8a;  ^  .  ^        3-lOa; 

3.i?  =  5,  5= -3;  ,     r:^.Q^2  4.i?  =  4,  ^=-5; 


l_5a;  +  3«2  -./---,«-     "'  i_4a;+5a;2 


486  ADVANCED  ALGEBRA. 

TOO  o  1—x—^x^ 

^  o  o  A  l-5x  +  Sx^ 

„          ^          ^     l-x-1809x^-Udlx-' 
'J'P  =  ^^9  =  ^;  1_3^_3^. 

8.  p  =  2,  ^  =  — 3 ; 


9.p=-2,  q  =  2; 


l-2ic  +  3a;2 
2  +  3a:-2208a;«  +  1616ic9 


10.  i)=l,  q=-l,  r=l; 


1— a;  +  a;2— ic* 


Exercise  95. 

,       2            3  „        3  2 

1.  — TR  +  7i  2. 


a:+2       a;-2  2a;  +  l       2a;-l 

,12  3  ^3  5 

3.    -  +    -— r  +  — ^  4. 


X       x  +  1       x  +  2  x  +  1       (x  +  l)^ 

15  7  -      3  4 


°'   ^J_.q  /'^_La\2    "'"    /^j..q\3  ^'   <rJ_9.   "^ 


a:+3       (a;+3)2       (a;  +  3)3  iC+2   '   (x-df 


P       ,         Q 


7.  +  8.  — ^- —  +  

a—x      o+x  px  +  q      (px+q)^ 

^2  3  ,^222 


x^  +  x  +  1      x^—x  +  1  X       l  +  2a;      1— 2a; 

5 7_  1  3 

(rc-l)*       (a;-l)3  "^  (ar-l)^  "^  a:-l 

12         ^       4.   ^-^^  13    1         3     4.      » 


1  a;+l  a;+5  ,,         1  x  +  l_ 

1) 


^^'       4(a;-l)  "^  4(a;«  +  l)  "*"  2(a;«  +  l)«  ^^"  2(a;+l)  "^  2(a;2  + 


,^111             1  9  1  x+1 

16.  -  -  +  -^ .  +  jt; TT  + 


X    '   x^       x^  '   8(a;-l)   '   8(a;  +  l)       4(a;  +  l)«       4(a;2  +  l) 
3a:  1         ,„       1  1  3  3  2 

17.   -o-t; = K        18.    r -:  +  ^—-^  -  .      ■  ..o  + 


a:«  +  2ic-5      a;-3  ic-l      ic  +  1      (a;  +  l)«      (a;  +  l)2     (a:+l)* 

,^111  1  9  1  a;  +  l 

19. +  -J 8  + 


a;       a:»      a:»  -    8(2;-l)       8(2;  +  l)       4(a;  +  l)»       4(a;»  +  l) 


1  x  +  1 

^""  3(a;  +  l)  "^2(0:2  +  1)     ^^' 

22. 


^iV^6'Tf^i2;S^. 

487 

3 

3 

"•"SCaj  +  l) 

1    "^-^ 

•    4:{X-lf 

8(a;-l) 

4(0:2+1) 

8 

13 

(x  +  1)       6  (a; +  1)2  "  45(a:-2)       20  (a; +  3) 

Exercise  97. 

1.  dy  =  {15ax^-6bx  +  2c)dx 

2.  dy  =  15z^x^dx  +  10x^zdz  +  dz  3.  <?y  =  (3a;2  +  6a;  +  4)fZa; 

Q 

4.  f?3/  =  5m2(a+Z>a;)"»''-icZa;  6.  (?2/  =  (a:+  -^  — a;-2)f?a; 

^    ,        ad^a;           1                        „     ,        5dx           1 
6,dy  —  — J- . r  1,  dy=-^-  .  ^ 

2ci      (aa;  +  6)t  3       (5aj  +  6)f 

8.  <Zy  =  /i/— .  <Za;  9,  dyz=z—i^a^)  .  x—^dx 

10.  dy= .  — ,  11.  dy  =  (— ax^+bx—i]dx 

a      ^a^-x^  ^      V2  ; 

,^     ,        2a;3(4a+a;)      ,  ,_     ,         i^x+a)dx 

12.  dy= — -T^ — ~-  .  dx  13.  dy  =  - -tt- 

^  («+^)*  ^         (a;+a)l 

14.  cZyzz:-^^^^^  15.  dy  =  {x+a){x+})f{^x+Za  +  h)dx 

(6'+a;2) 

16.  <?2/  =  (a;  +  a)»-»(a;-6)i»-»(2a;+a-&)(?a;        17.  dy-"^^^  .  da; 

o  ^  ^ 

18.  6?2/  =  (logea:)2  . 19.  dy  =  9iQ(?^{^  +  \og^Z  +  \oz^x)dx 

X 

20.  dy  =  2a»*.a;logea<ia;  21.  dy=z- ^(2a  +  3a;2) 

(a+a;2)t. 

22.  dy  =  ^-j-|  23.  dy-  (^^Y.  (loge  c-loge  d)dx 

«^     7  2<?a;  «,^     ,  ,  ,  dx 

(l_a;)(l-a;2)i  ^  ^'  a; 

Exercise  98. 

2.  16  sq.  in.  per  second.  3.  4'irr2;  144  ir  cu.  in.  per  second. 

4.  2  V' 2  in.  5.  1  in.  per  second.        6.  About  6  mi.  an  hr. 
7.  12  V^  in.  per  sec.     8.  -00517  9.  1-62842 ;  -00003 

Exercise  99. 

1.  3a;2-8a;  +  7  2.  (a;+2)2(a;-2)3(7a;  +  2) 

3.  3a:2  +  8a;+2  4.  (a+a;)4(a-a:)2(8a-2a;) 

5.  -8a:'  +  5aa:*-3aa;2  6.  -10a;  .  ^^^ 

(a—xf 
p 


488  ADVANCED  ALGEBRA. 

Exercise  100. 

1.  {x  +  Zf{x-2f  2.  (x-2)^{x-iy(x+d) 

3.  {x  +  Sf(x-3f(x^+x+l)  4.  (x-2)\x  +  2f{x-df(x+Sy 

5.  {x-lY(x+iy{x+d)(x-d) 

Exercise  101. 

Q 

1.  Max.  1  -J ;  min.  —5  2.  Min.  —3 ;  max.  —128 
3.  Min.  -4         4.  Min.  -4  5.  Max.  10 ;  min.  -22 
6.  Min.  0;  max.  18+                        7.  Max.  (^Y 

8.  No  turning  values.  9.  Min.  —14  10.  Min.  —14 

11.  Min.  0  12.  Min.  —16        13.  At  the  middle  point. 

1  32  8 

15.  o-a  16.  ^  V  r*,  or  ^r^  of  the  vol.  of  the  sphere. 

O  ol  at 

17.  -^a^  or  an  inscribed  square.     18.  -j  <ir  a^  19. -^  a 

20.  Square  =  2r'  21.  An  isosceles  triangle. 

Exercise  102. 

1.x* +  2 a^-9 x^  +  63 a;-135  =  0 ;  roots  of  fn {x)  =  dx  roots  of  Fn (x) 

2.  x^i-4: x*—2 x^+4 a;2— 112  =  0 ;  roots  of  /« {x)  =  2x  roots  of  Fn (x) 

3.  x^  + 12  a;*-320  x^  + 1792  a;- 1024  =  0 ;  roots  of  /„  (x)  = 

4x  roots  of  ^n(a;) 

4.  a;'^-2a;*  +  9a;3-18a;2  +  108a;-162  =  0;  roots  of /„ (a:)  = 

3  V^oots  of  (A) 

5.  a:«-2  a;5  +  2  x*-4:  a:* + 24  a:^  +  32  a;  +  32  =  0 ;  roots  =  2  Vroots  of  {A) 

6.  a;3«  +  8  .  9^x^+36  .  9»a;24  +  10  .  9^9a;'«-4  .934  =  0;  roots  = 

9  Vroots  of  {A) 


7.  a;''-a;6  +  18a:*-162a;9-2187  =  0;  roots  =  3  Vroots  of  {A) 

8.  a;«-10a;*-36a:4_i2288  =  0;  roots  =  4  Vroots  of  (^) 

9.  a;4  +  6800a;-9000  =  0 ;  roots  =  10  Vroots  of  (A) 


Exercise  103. 

1.  a;  =  l                2.  a;  = 
5.  a;  =  1,  2,  2,   -  2 

:  4                3.  a;  =  5 

6.a;  =  i,   ■ 

— 

4.  a;  = 
1           2 
2'        3 

7.  a;  =  l,  2,   -2,   -3 

8.  a;  =  5,   - 

■1 

9.  a;=  +2,   -2,   -3 

10.  a:  =  2,  4 

-2 


ANSWUES.  489 

13.  ic  =  3,   -2,   -1  14.  a:=-3,   -5,   -1,    ±| 

15.  a:  =  -l,  ll,   -li  16.  ic  =  2,   -3,   -5,  2 

17.  re  =  2,  2,   -3,   -3  18.  a;  =  1,  1,   -2,   -2,   -2 

19.  a;  =  -1,   -1,    -1,  2,   -3,   -| 

Exercise  104. 

1.  -1  +  ,  4  +  ,  -9+                         2.  -6-6+  3.  0-3  + 

4.  0-8  +  ,  3  +  ,  -3  +  ,   -0-1  +  ,   -0-6  + 

5.  2  +  ,   -2+                                    6.  2  +  ,  0-6  +  ,  0-4  +  ,   -3  + 

Exercise  105. 

1.  775  2.  240  3.  -87938,   -2*53208,   -1-34729 

4.  3-85808,   1-60601,  1-44327,   -2-90737 

5.  -5,  4-05607,  11-15306,   12-79085 

6.  -1-12579         7.  2-34244  8.  1-25992  9.  1-3797 
10.  3-2131,  3-2295,   -17-4426          11.  -80285,   -5-4335 

12.  8-41445  13.  a;3  4.32a;2  +  343a;-13087800  =  0 

Exercise  106. 

1.  a;  =  l,  1±2\/^  2.  a;  =  4,  1±\/^ 

3.a;  =  5,  3±\/^  4,x=-2±^/^ 

6,x  =  d,  i(l±V^  6,x=-4,   _i(l±v^ 

1,  x=  -2,   -2,   -3  8.  x  =  3,   -2,   -2 

9.  a:  =  2,   2,   -3  10.  a;  =  4,   -2±'v/-^ 

1 
2 

13.  a;  =  5,  1,   -2  14.  a:  =-5,   -1,  2 


11.  a;=r-6,  3±'\/^  12.  a;  =  1,    ^  (3±  V^l) 


Exercise  107. 

1.  a;  =  l,  i(l±^Z3")  2.  a^  =  -l,  2±\^ 

Z.x  =  \,   -1,   i(3±V5)  4.  a:  =  l,   -1,   \{-^±V~^) 

5.  a;  =  i(5±V2T,    ±a/^  6.  a;  =  -l,   -1,   -1,   -1,    -1 

7.  a;  =  l,  2,  i,   -3,   -i  8.  a;  =  -l,   -5,   -1,  2±V3 


490 


ADVANCED  ALGEBRA, 


Exercise  108. 

1.  -9 

2.  -59                 3.  65 

4.  o&c— 2a6* 

5.  ^mnp 

6.  4:abc               7.  28 

8.  -82 

9.  -108 

10.  0                      11.  0 

12.  0 

13.  0 

14.  0                      15.  0 

Exercise  109. 

1.  0 

2.  0                       3.  0 

4.  0 

5.  Sxyz 

6.  6a&c-2a3_2  63_2c3 

1.  10 


Exercise  110. 

-48  3.  -199 


4.  0 


1.  X 

3.  X 

4.  X- 

6.  X: 

7.  X  ■ 


Exercise  111. 

2.  a;  =  3,  y  =  S 


y 

z 

9.  X: 


4,  y  =  2 

cn—hd  _  ad—c  m 
an—bm^  ^~an—bm 
cm—dm  +  cn  +  dn       _  an  +  bn—am  +  bm 

2{ac-bd)        '  ^~         2iac-bd) 
2,  2/  =  3,  2  =  4  6.  x  =  5,  y  =  l,  2  =  '. 

abd—acd—abe  +  ec^  +  abh—bc  h 

a^  b—a^  c—a  b  c  +  c^  +  a  b^—b^  c 
a^  e—a^  h—a  cd+c^h  +  ab  d—b  c  e 

aH—a^c—abc  +  c^  +  ab^—bU 
abh—ace—bch  +  c^d  +  b^e—b^d 
a^  b—a^  c—a  bc  +  c^+a  b^—¥  c 
_  a^(m  +  n—r)  +ab(m—n—r)  +b^(7n—n  +  r) 
""  2aF+2¥ 

_  a^(n—m  +  r)  +ab{2an—r—m—n)  +b^(n—r+m) 
—  2a3  +  2  63 

a^{r—n-k-m)  +ab(2r—n—r—m)  +b^(r+n—m) 
■  2aF+2¥ 

2,  3/  =  3,  2  =  4,  u  =  5 


y  = 


z=. 


10.  x  = 


c  0  a 

b  c  0 

b  c  0 

b  c  0 

m 

0  a  b 

—n 

0  ab 

+p 

c  0  a 

-Q 

c  0  a 

a  b  c 

a  b  c 

a  b  c 

0  ab 

c  0  a 

b  c  0 

b  c  0 

a 

0  ab 

-b 

0  ab 

+  c 

c  0  a 

a  b  c 

a  b  c 

a  b  c 

ANSWURS. 


491 


y  =  -{a 


11.  a;  =  2,  _ 

_p  +  q—r—s 


n  0  a 

m  c  0 

m  c  0 

pah 

-b 

p   a  b 

+  c 

71  0  a 

q  b  c 

q    b  c 

q    b  c 

c  n  a 

b  m  0 

b  m  0 

Op  b 

-b 

Op   b 

+  c 

c  n  a 

a  q  c 

a  q  c 

a  q  c 

c  0  n 

b  c  m 

b  c  m 

0  a  p 

-b 

0  a  p 

+  c 

c  0  n 

a  b  q 

a  b  q 

a  b  q 

=  3,  Z-- 

=  -3, 

w=-3 

p—q  +  r—8 

4& 
p—q—r+8 
Ac        '  '""         4d 
13.  a;  =  1,  J/ =  2,  2  =  3,  u  =  4,  v  =  5 


u  = 


\- 


-^c.d 


-i-  c  .  d 


1.  Consistent. 


Exercise  112. 

2.  Inconsistent. 


i  1  1  i  i  i 

2+1+1+6+2+4 

111 


Exercise  115. 

1     1 


2. 


_  _  i  1  i 

1  +  1  +  1+11  +  6 


3.  ^  .-=^ 


6.  3  + 


1 


+  1  +  99 
i      i 


4   i    i    1    i    i 

6+1+7+6+2 


5.  3  + 


2  +  6 
1       1 


7.  5  + 


1^     ]_ 

2  +  10 


8.  7+  ^  .  4 


!_     1^ 

1  +  4 


1     1 


3  +  7  +  17  +  2  +  1  +  8 
11-67+^^^^^^^^^^deg. 


10.  3  + 


12. 


i     J_     1 

7+16  +  11 

3     13      68 


2'    7'  80'  157 


1^     1^     10     11 

3'    4'  39'  43 

1      2      7      16 

23 


13. 
15. 
17. 
21. 
23.  4,   4 


14.^, 


1  '    3'   10' 

\/5-2 

2  5     J^ 
3'    9'   12 

1  1     15 

2  '   3  '  29 


55 
79' 


etc. 
18.  V6-2 


16.  2  ,  ^, 
19.  VS 


16' 
10 
23 


24     55     189 

55'  126'  433 

33     109 

'   76'  251 


etc. 


etc. 


20.  i(VlO-l) 


19     26 
33'  45 
21 
61 


123 


etc. 


22. 


24. 


492  ADVANCED  ALGEBRA. 


Esercise   116. 

1.  25542               2.  616 

3.  434005, 

327454 

4.  58072 1,  32853:^ 

5.  2950 

6.  125344 

7.  2e23                8.  202033030 

9.  7692 

10.  14860 

11.  r  =  6               12.  r=6 

13.  r  =  14 

14.  2 

15.  1414,  1201 

16.  r  =  7 

Exercise  117. 

1.  197,  251,  313,  281,  461,  829     2.  6       3.  1646 
4.  By  9:  11205,  342738,  558657 

By  11 :  24530,  342738,  25916,  558657 

By  9  and  11 :  342738,  558657 

1.  x=^  8.  a:  =  ^ 


THE  END. 


APPLETONS' 
MATHEMATICAL    SERIES, 


FOUR    VOLUMES. 


Beautifully  Illustrated.    The  Objective  Method 
Practically  Applied. 


THE    SERIES: 

I.    Numbers  Illustrated 

And  applied  in  Language,  Drawing,  and  Reading  Lessons. 
An  Arithmetic  for  Primary  Schools.     By  ANDREW   J. 
RICKOFF,  LL.  D.,  and  E.  C.  DAVIS. 
Introduction  price,  36  cents. 

II.    Numbers  Applied. 

A  Complete  Arithmetic  for  all  Grades.    Prepared  on  the 
Inductive  Method,  with  many  new  and  especially  practical 
features.     By  ANDREW  J.  RICKOFF,  LL.  D. 
Introduction  price,  75  cents. 

III.  Numibers  Synribolized. 

An  Elementary  Algebra.  By  DAVID  M.  SENSENIG,  M.  S., 
Professor  of  Mathematics  in  the  State  Normal  School  at 
West  Chester,  Pa. 

Without  Answers — Introduction  price,  $1.08. 
Witli  Answers— Introduction  price,  $1.16. 

IV.  Numbers  Universalized. 

An  Advanced  Algebra.     By  DAVID  M.  SENSENIG,  M.  S. 


These  books  are  the  result  of  extended  research,  as  to  the  best  methods 
now  in  use,  and  many  years'  practical  experience  in  class-room  work  and 
school  supervision. 

Send  for  full  descriptive  circular.  Specimen  copies  will  be  mailed  to 
teacJiers  at  the  introduction  prices. 


D.  APPLETON  &  CO.,  Pcblis 

New  Yokk,  Boston,  Chicago,  Atlanta,  San  Fbanoisoo. 


HIGHER  MATHEMATICS, 


Elements  of  Geometry.  By  Eli  T.  Tappan,  LL.  D.,  Professor 
of  Political  Science  in  Kenyon  College,  formerly  Professor  of 
Mathematics.     12mo,  253  pages. 

Introductory  price,  92  cents. 

This  work  lifts  Geometry  out  of  its  degraded  position  as  mere  intellectual  gym- 
nastics. The  author  holds  that  certain  knowledge  of  the  truth  we  begin  with  is  as 
important  as  the  process  of  inference,  and  he  has  aimed,  first,  to  state  correctly  the 
principles  of  the  science,  and  then,  upon  these  premises,  to  demonstrate,  rigorously 
and  in  good  English,  the  whole  doctrine  of  Elementary  Geometry,  developing  the  Bub- 
jeet  by  easy  gradations  from  the  simple  to  the  complex. 

Elements  of  Plane  and  Spherical  Trigonometry, 

with  Applications.     By  Eugene  L.  Richards,  B.  A.,  Assistant 
Professor  of  Mathematics  in  Yale  College.     12mo,  295  pages. 

Introductory  price,  91. /30. 

The  author  has  aimed  to  make  the  subject  of  Trigonometry  plain  to  beginners,  and 
much  space,  therefore,  is  devoted  to  elementary  definitions  and  their  applicati(<n8.  A 
free  use  of  diagrams  is  made  to  convey  to  the  student  a  clear  idea  of  relations  of  mag- 
nitudes, and  all  difiScult  points  are  fuUy  explained  and  illustrated. 

Thk  Same,  with  Tables. 

Introductory  price,  $1.50. 

Williamson's  Integral  Calculus,  containing  Applications 

to  Plane  Curves  and  Surfaces,  with  numerous  Examples.     12mo, 
375  pages. 

Introductory  price,  83.00. 

Williamson's  Differential  Calculus,  containing  the  The- 
ory of  Plane  Curves,  with  numerous  Examples.    12mo,  416  pages. 

Introductory  price,  83.00. 


Sample  copies,  for  examination,  will  be  mailed,  post-paid,  to  teachers^ 
at  the  above  introductory  prices.  Send  for  fvXl  descriptive  list  of  texU 
book*  for  all  grades. 


D.  APPLETON  &  CO.,  PUBLISHERS, 
New  York,  Boston,  Chicagro,  Atlanta,  San  Francisco. 


A  TREATISE  ON  SURVEYING; 

COMPRISING  THE  THEORY  AND  THE  PRACTICE. 

BY 

William  M.  Gillespie,  LL.  D., 

Formerly  Professor  of  Civil  Engineering  in  Union  College. 
Eevlsed  and  Enlarged  by 

Cady  Staley,  Ph.  D., 
President  qf  the  Case  School  of  Applied  Science. 


1  vol.,  8vo,  549  pages.  Folly  Illustrated,  New  Plates,  eto. 


A  complete  and  systematic  work  covering  the  whole  subject  of  prac- 
tical and  theoretical  surveying,  embodying  in  one  volume  Gillespie's 
"Land  Surveying"  and  "Leveling  and  Higher  Surveying,"  which  for 
years  have  been  acknowledged  authorities  on  the  subject.  These  two 
works  have  been  thoroughly  revised  by  Professor  Staley  and  united  in  a 
single  volume — especially  adapted  for  class  use  in  high-schools  and  col- 
leges. 

In  the  preparation  of  this  combined  volume  a  double  object  has  been 
achieved  ;  it  is  not  only  a  comprehensive  course  for  scholars  desiring  a 
practical  knowledge  of  the  subject — without  thought  of  making  it  a  life 
work — but  it  also  lays  a  foundation  deep  enough  and  broad  enough  for 
the  most  complete  superstructure  which  the  professional  student  may 
wish  to  raise  upon  it. 

The  work  includes  Land  Surveying,  Leveling,  Topography,  Triangular 
Surveying,  Hydrographical  Surveying,  and  Underground  or  Mining  Sur- 
veying, with  valuable  appendices  on  Plane  Trigonometry,  Transversals, 
etc. ;  and  full  sets  of  Tables. 


Sample  copies  will  be  forwarded^  post-paid^  to  teachers^  for  ezaminci- 
tion,  on  receipt  of  the  introductory  price — $3.00. 


D.  APPLETON  &  CO.,  PUBLISHERS, 
Kew  York,  Boston,  Chica^Ot  Atlanta,  San  Francisco. 


STANDARD  TEXT-BOOKS, 


Appletons'  Readers. 

SIX  BOOKS.  Perfectly  graded,  beautifully  illustrated.  These  books 
have  held  a  foremost  place  among  school  readers  from  the  first  day  of 
their  publication  to  the  present  time,  and  they  will  continue  for  many 
years  to  delight  the  hearts  of  thousands  of  children,  who  will  ever  find 
new  pleasure  in  their  freshness  and  novelty. 

' '  Al  -ways  ne  w."      "  Al  ways  in  teresting." 

Appletons'  Standard  Geographies, 

ELEMENTARY,  HIGHER,  PHYSICAL.  Unequaled  in 
point,  attractiveness,  and  completeness.  Thoroughly  up  to  date  in  all 
departments.  The  new  Physical  Geography  was  prepared  by  a 
corps  of  scientific  specialists,  presenting  an  array  of  talent  never  before 
united  in  the  making  of  a  single  text-book.  It  stands  unrivaled  among 
works  on  the  subject. 

Appletons'  Mathematical  Series. 

NUMBERS  ILLUSTRATED.    By  A.  J.  Eickoff  and  E.  C. 

Davis. 
NUMBERS  APPLIED.    By  A.  J.  Eickoff. 
NUMBERS  SYMBOLIZED.    By  D.  M.  Sensenig. 
NUMBERS  UNIVERSALIZED.    By  D.  M.  Sensenig. 

The  *'  objective  method"  successfully  applied.    A  distinct  advance 
on  any  mathematical  works  heretofore  published. 

Appletons'  Standard  System  of  Penmanship, 

Perfectly  adapted  for  all  grades.  The  only  books  in  which  graded 
columns  are  used  to  develop  movement. 

Krusi's  System  of  Drawing. 

FREE-HAND,    INVENTIVE,    INDUSTRIAL.       For  all 

grades.    Strictly  progressive.     Thoroughly  educational. 

Introductory  Course.      Supplementary  Course. 
Graded  Course.  Industrial  Courses. 


Send  for  full  descriptive  circvlarSy  terms  for  introduction  y  etc. 

D.  APPIiETON  A  CO.,  Publishers, 

New  Yorhj  JSodon,  Chicago^  Atlanta^  San  Francisco, 


APPLETONS'  SCIENCE  TEXT-BOOKS. 


In  response  to  the  growing  interest  in  the  study  of  the  Natural  Sci- 
ences, and  a  demand  for  improved  text-books  representing  the  more 
accurate  phases  of  scientific  knowledge,  and  the  present  active  and 
widening  field  of  investigation,  arrangements  have  been  made  for  the 
publication  of  a  series  of  text-books  to  cover  the  whole  field  of  science- 
study  in  High  Schools,  Academies,  and  all  schools  of  similar  grade. 

The  following  are  now  ready.     Others  in  preparation. 

THE  ELEMENTS  OF  CHEMISTRY.  By  Professor  F.  W. 
Clarke,  Chemist  of  the  United  States  Geological  Survey.  12mo, 
369  pages. 

THE  ESSENTIALS  OF  ANATOMY,  PHYSIOLOGY, 
AND  HYGIENE.  By  Roger  S.  Tracy,  M.  D.,  Sanitary  In- 
spector  of  the  New  York  Board  of  Health.     12mo,  299  pages. 

ELEMENTARY  ZOOLOGY.  By  C.  F.  Holder,  Fellow  of  the 
New  York  Academy  of  Science,  Corresponding  Member  Linnaean 
Society,  etc. ;  and  J.  B.  Holder,  M.  D.,  Curator  of  Zoology  of 
American  Museum  of  Natural  History,  Central  Park,  New  York. 
12mo,  385  pages. 

A  COMPEND  OF  GEOLOGY.  By  Joseph  Le  Conte,  Professor 
of  Geology  and  Natural  History  in  the  University  of  California; 
author  of  "Elements  of  Geology,"  etc.     12mo,  399  pages. 

APPLIED  GEOLOGY.  A  Treatise  on  the  Industrial  Relations  of 
Geological  Structure,  By  Samuel  G.  Williams,  Professor  of  Gen- 
eral and  Economic  Geology  in  Cornell  University.    12mo,  386  pages. 

DESCRIPTIVE  BOTANY.  A  Practical  Guide  to  the  Classifi- 
cation  of  Plants,  with  a  Popular  Flora.  By  Eliza  A.  Youmans. 
12mo,  336  pages. 

PHYSIOLOGICAL  BOTANY.  By  Robert  Bentley,  F.  L.  S., 
Professor  of  Botany  in  King's  College,  London.  Adapted  to  Ameri- 
can Schools  and  prepared  as  a  Sequel  to  "  Descriptive  Botany,"  by 
Eliza  A.  Youmans.     12mo,  292  pages. 

THE   ELEMENTS   OF  POLITICAL  ECONOMY.     By  J. 

Laurence  Laughlin,  Ph.  D.,  Assistant  Professor  of  Political  Econ- 
omy in  Harvard  University.     12mo. 


For  Bpecimen  co|)ieB,  terms  for  introdnction,  catalogue,  and  price-list  of  all 
our  pubhcations,  write  to  publishers  at  either  address  below. 

D.  APPLETON  &  CO.,  Publishers, 

NEW  YORK,   BOSTON,   CHICAGO,  ATLANTA,   SAN    FRANCISCO. 


EGGLESTON'S  AMERICAN  HISTORIES. 


A  First  Book  in  Ameriean  History. 

WITH  SPECIAL  REFERENCE  TO  THE  LIVES 
AND  DEEDS  OF  GREAT  AMERICANS.  By  Edward 
Eggleston.  This  is  a  history  for  beginners  on  a  new  plan. 
It  makes  history  delightful  to  younger  pupils,  by  introducing 
them  to  men  who  are  the  great  landmarks  of  our  country's 
story.  The  book  comprises  a  series  of  biographical  sketches 
of  more  than  a  score  of  men  eminent  in  different  periods  of 
American  history.  Beautifully  illustrated  by  the  most  emi- 
nent American  artists.    Introduction  price,  60  cents. 

A  History  of  the  United  States  and 
its  People. 

FOR  THE  USE  OP  SCHOOLS.  By  Edwaed  Eggleston. 
Introduction  price,  $1.05. 


From  Eon.  LEWIS  MILLER,  Akron,  Ohio. 

"  I  have  looked  over  the  History  and  like  it  very  much." 


From  Bishop  JOHN  H.  VINCENT,  D.D.,  LL.  D., 

Chancellor  of  Chautauqua  University. 

"I  regard  this  beautiful  volume  as  the  "highest  standard  of  school- 
book  yet  attained." 

From  W.  B.  POWELL,  Superintendent  of  Schools,  Washington,  D.  C. 

"  Our  teachers  en  masse,  and  tliousands  of  our  pupils,  are  delighted 
with  your  American  History." 


Specimen  pages,  terms  for  introduction,  etc.,  will  be  forwarded 
on  application. 

D.  APPLETON  &  CO.,  Publishers, 

New  York,  Boston,  Chicago,  Atz.anta,  San  Fbanoisoo. 


THIS  BOOK  IS  DUE  ON  THE  LAST  DATE 
STAMPED  BELOW 


AN  INITIAL  FINE  OF  25  CENTS 

WILL  BE  ASSESSED  FOR  FAILURE  TO  RETTURN 
THIS  BOOK  ON  THE  DATE  DUE.  THE  PENALTY 
WILL  INCREASE  TO  50  CENTS  ON  THE  FOURTH 
DAY  AND  TO  $1.00  ON  THE  SEVENTH  DAY 
OVERDUE. 


FEB 


FED    231M4 

r)"^/ 

/             { 

APR  2  8  19$g  6  9 

1 

^•,'r;^,-,'     ^- 

LD  21-1007n-7,'33 

A 


f88r?8| 


